Chirality in liquid crystals

Partially Ordered Systems Editor in Chief: L. Lam San Jose State University San Jose, California, USA Editorial Board: E. Guyon EÂcole Normale SupeÂrieure Paris, France

D. Langevin Laboratoire de Physique ENS Paris, France

H.E. Stanley Boston University Boston, Massachusetts, USA Advisory Board: J. Charvolin Institut Laue-Langevin Grenoble, France

W. Helfrich Freie UniversitaÈt Berlin, Germany

P.A. Lee Massachusetts Institute of Technology Cambridge, Massachusetts, USA

J.D. Litster Massachusetts Institute of Technology Cambridge, Massachusetts, USA

D.R. Nelson Harvard University Cambridge, Massachusetts, USA

M. Schadt ROLIC Research Ltd Allschwil, Switzerland

This page intentionally left blank

Heinz-Siegfried Kitzerow Editors

Christian Bahr

Chirality in Liquid Crystals Foreword by Sivaramakrishna Chandrasekhar With 326 Illustrations

Heinz-Siegfried Kitzerow Department of Chemistry University of Paderborn Warburger Strasse 100 D-33098 Paderborn Germany [email protected]

Christian Bahr Institute of Physical Chemistry University of Marburg Hans-Meerwein-Strasse D-35032 Marburg Germany [email protected]

Editorial Board: Lui Lam Department of Physics San Jose State College One Washington Square San Jose, CA 95192 USA

Dominique Langevin Laboratoire de Physique des Solides Batiment 510 Universite Paris Sud F-91405 Orsay France

Etienne M. Guyon EÂcole Normale SupeÂrieure 45 Rue D'Ulm F-75005 Paris France

H. Eugene Stanley Center For Polymer Studies Physics Department Boston University Boston, MA 02215 USA

Library of Congress Cataloging-in-Publication Data Chirality in liquid crystals / editors, Heinz-Siegfried Kitzerow, Christian Bahr. p. cm. Ð (Partially ordered systems) Includes bibliographical references and index. ISBN 0-387-98679-0 (hard cover : alk. paper) 1. Liquid crystals. 2. Chirality. I. Kitzerow, Heinz-Siegfried. II. Bahr, Christian. III. Series. QD923.C55 2001 541 0 .04229Ðdc21 99-052790 Printed on acid free paper. ( 2001 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 1010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identi®ed, is not to be taken as a sign that such names as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Michael Koy; manufacturing supervised by Jerome Basma. Typeset by Asco Typesetters, Hong Kong. Printed and bound by Maple-Vail Book Manufacturing Group, York, PA. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 ISBN 0-387-98679-0

SPIN 10700911

Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science‡Business Media GmbH

This book is dedicated to Prof. Gerd Heppke with respect and kind regards.

This page intentionally left blank

Foreword

It is indeed a pleasure for me to write the foreword for this volume brought out in honor of Prof. Gerd Heppke to felicitate him on his reaching sixty years of age. It comprises a collection of papers by friends and admirers on the broad theme of chirality in liquid crystals, which is an area of current interest and one in which Prof. Heppke himself has made signi®cant contributions. A Berliner by birth, Prof. Heppke has been associated with the Technical University Berlin for over thirty-®ve years in various capacities. After his Ph.D. and Habilitation, he became, successively, Assistant Professor (1972), Professor of Physical Chemistry (1982), Chairman of the University Research Area ``Liquid Crystals and Their Electrooptic Applications'' (1981±86), and Vice-Chairman of the Special Research Area ``Anisotropic Fluids,'' supported by the Deutsche Forschungsgemeinschaft (1987±98). During this period he played a vital role in laying the foundation for a strong interdisciplinary thrust area involving some twenty research projects from di¨erent laboratories. Prof. Heppke is an editor of the journal Molecular Crystals and Liquid Crystals and is a nonexecutive director of the International Liquid Crystal Society. His research interests cover diverse aspects of liquid crystal science, as can be judged from his many publications (nearly two hundred) on the subject. The range of his contributions include: the polymorphic forms of smectic phases and the topology of their phase diagrams, critical and multicritical points, reentrant nematics, guest-host e¨ect, dual frequency addressable mixtures, optical storage e¨ects, determination of the screw sense of cholesterics, electric ®eld e¨ects in blue phases, chiral dopants with extremely high helical twisting power, ferroelectrics with very high spontaneous polarization, antiferroelectric phases, electroclinic e¨ect, electromechanical e¨ect, electrooptic e¨ect in nematic discotics, selective re¯exion and blue phases in chiral discotics, ferroelectric columnar phases, banana shaped molecules, etc. He has organized several international conferences on liquid crystals and edited their proceedings. As a teacher and research guide, he has been the supervisor of numerous diploma and Ph.D. research students, several of whom have been recipients of awards and prizes for their excellent work vii

viii

Foreword

(for example, one of his students, Dr. Christian Bahr, received the Glenn H. Brown Award for the synthesis and study of ferroelectric compounds exhibiting very high spontaneous polarization). Apart from his own personal research interests, one of Prof. Heppke's major achievements has been the part he has played in initiating international exchange between his team at the Technical University Berlin and other liquid crystal groups throughout the world. Over the years, the TU group has had active collaboration and scienti®c exchange with the Raman Research Institute, Bangalore, India; the Universite Paris-Sud at Orsay; the Universite Montpellier II; the Massachussetts Institute of Technology; the University of Jammu, India; the Naval Research Laboratory, Washington DC; the University of Hawaii, USA; the University of Goteborg, Sweden; the University of Hull, UK; and research centers at Moscow, Tbilisi, and other places. It would be no exaggeration to say that all those who participated in this program were without exception very much impressed by the kindness and hospitality extended to them by Prof. Heppke and his coworkers, and they also found the visits to be extremely useful scienti®cally. Finally, it remains for me to express on behalf of all the contributors to this volume, colleagues and friends, our warmest greetings to Gerd Heppke on the occasion of his sixtieth birthday and to wish him many more fruitful and active years ahead. Sivaramakrishna Chandrasekhar Centre for Liquid Crystal Research Bangalore, India

Preface

The existence or nonexistence of mirror symmetry plays an important role in nature. The lack of mirror symmetry, called chirality, can be found in systems of all length scales, from elementary particles to macroscopic systems. Due to the collective behavior of the molecules in liquid crystals, molecular chirality has a particularly remarkable in¯uence on the macroscopic physical properties of these systems. Probably, even the ®rst observations of thermotropic liquid crystals by Planer (1861) and Reinitzer (1888) were due to the conspicuous selective re¯ection of the helical structure that occurs in chiral liquid crystals. Many physical properties of liquid crystals depend on chirality, e.g., certain linear and nonlinear optical properties, the occurrence of ferro-, ferri-, antiferro- and piezo-electric behavior, the electroclinic e¨ect, and even the appearance of new phases. In addition, the majority of optical applications of liquid crystals is due to chiral structures, namely the thermochromic e¨ect of cholesteric liquid crystals, the rotation of the plane of polarization in twisted nematic liquid crystal displays, and the ferroelectric and antiferroelectric switching of smectic liquid crystals. The intention of this book is to give an overview of the main aspects of chirality in liquid crystals, and to point out some of the open questions of current research. A complete description of this important subject within one volume is hardly possible. Thus, we have asked some experts to give a review of their ®eld of interest rather than collecting all aspects in a lexical manner. We hope that the following chapters give a representative impression of the interesting questions that are being investigated in the ®eld of chiral liquid crystals, even if some pieces are missing. We thank all authors who contributed to this book for their pleasant cooperation, and are very grateful to Prof. Chandrasekhar for his kindness in writing the foreword. Thomas von Foerster, Jeannette Mallozzi, Michael Koy, Keisha Franklin, Jerome Basma, and Springer-Verlag deserve our thanks for their valuable collaboration in publishing this volume. Gratefully, we appreciate the support by our colleagues who have agreed to the reproduction of previously published data, and would like to thank all those who did not hesitate to send us photographs and diagrams for the purpose ix

x

Preface

of reproduction. We express our thanks to Mrs. Marlies Kensbock and Mrs. Isabella Koralewicz, who helped considerably in the editing process by keeping track of the manuscripts, assisting in the correspondence, and writing the table of contents. In addition, we thank Mrs. Gisela JuÈnnemann, Mrs. Susanne Keuker-Baumann, and Mrs. Claudia Stehr for their assistance in searching articles from the literature and drawing some of the ®gures. Gratefully, we acknowledge Dr. Janusch Partyka's help, who assisted in the handling of computer ®les and gave some valuable hints on typographical errors in the ®nal manuscript. Prof. Gerd Heppke, our teacher and former adviser, has made numerous contributions to the ®eld of chiral liquid crystals. The great merits of his hitherto existing works are addressed in the foreword by Prof. Chandrasekhar. Thanks to Prof. Heppke's enthusiasm we are working in this ®eld, and thanks to his teaching and advice we were able to accumulate our knowledge of this interesting subject. On the occasion of his sixtieth birthday, we would like to dedicate this book to Prof. Heppke. On behalf of his current and former students and co-workers, we take this opportunity to express our gratitude and appreciation, our congratulations on his birthday, and best wishes for his forthcoming activities. Christian Bahr and Heinz-Siegfried Kitzerow

Contents

Foreword Sivaramakrishna Chandrasekhar

vii

Preface Christian Bahr and Heinz-Siegfried Kitzerow

ix

Contributors 1 Introduction Heinz-Siegfried Kitzerow and Christian Bahr

xiii 1

2 Classroom Experiments with Chiral Liquid Crystals Pawel Pieranski

28

3 From a Chiral Molecule to a Chiral Anisotropic Phase Hans-Georg Kuball and Tatiana HoÈfer

67

4 Chemical Structures and Polymorphism Volkmar Vill

101

5 Cholesteric Liquid Crystals: Defects and Topology Oleg D. Lavrentovich and Maurice Kleman

115

6 Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics Guram Chilaya 7 Blue Phases Peter P. Crooker

159 186

8 Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect Christian Bahr

223

9 Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases Hideo Takezoe and Yoichi Takanishi

251 xi

xii

Contents

10 Twist Grain Boundary Phases Heinz-Siegfried Kitzerow

296

11 Columnar Liquid Crystals Harald Bock

355

12 Some Aspects of Polyemer Dispersed and Polymer Stabilized Chiral Liquid Crystals Gregory P. Crawford, Daniel SvensÏek, and Slobodan ZÏumer

375

13 Chirality in Liquid Crystal Elastomers Peter Stein and Heino Finkelmann

433

14 Phase Chirality of Micellar Lyotropic Liquid Crystals Karl Hiltrop

447

15 Traveling Phase Boundaries with the Broken Symmetries of Life Patricia E. Cladis

481

Index

495

Contributors

Christian Bahr Institute of Physical Chemistry, University of Marburg, D-35032 Marburg, Germany Harald Bock Centre de Recherche Paul Pascal, F-33600 Pessac, France Sivaramakrishna Chandrasekhar (Foreword) Centre for Liquid Crystal Research, Bangalore, India Guram Chilaya Institute of Cybernetics, Academy of Sciences of Georgia, 380086 Tbilisi, Georgia Patricia E. Cladis Advanced Liquid Crystal Technologies, Inc., Summit, NJ 07902-1314, USA Gregory P. Crawford Division of Engineering, Brown University, Providence, Rhode Island 02912, USA Peter P. Crooker Dept. of Physics and Astronomy, University of Hawaii, Honolulu, HI 96822, USA Heino Finkelmann Institute of Macromolecular Chemistry, University of Freiburg, D-79104 Freiburg, Germany Karl Hiltrop Physikalische Chemie, UniversitaÈt-GH Paderborn, D-33098 Paderborn, Germany Tatiana HoÈfer Physikalische Chemie, UniversitaÈt Kaiserslautern, D-67663 Kaiserslautern, Germany Heinz-S. Kitzerow Department of Chemistry, University of Paderborn, D-33098 Paderborn, Germany xiii

xiv

Contributors

Maurice Kleman Laboratoire de Physique des Solides, Universite Paris-Sud, F-91405 Orsay, France Hans-Georg Kuball Lehrstuhl fuÈl Physikalische Chemie, UniversitaÈt Kaiserslautern, D-67663 Kaiserslautern, Germany Oleg D. Lavrentovich Chemical Physics Interdisciplinary Program, Liquid Crystal Institute, Kent State University, Kent, Ohio 44242-0001, USA Pawel Pieranski Laboratoire de Physique des Solides, Universite Paris-Sud, F-91405 Orsay, France Peter Stein Institute of Macromolecular Chemistry, University of Freiburg, D-79104 Freiburg, Germany Daniel SvensÏek Department of Physics, University of Ljubljana, 1000 Ljubljana, Slovenia Yoichi Takanishi Department of Organic and Polymeric Materials, Tokyo Institute of Technology, Tokyo 152-8552, Japan Hideo Takezoe Department of Organic and Polymeric Materials, Tokyo Institute of Technology, Tokyo 152-8552, Japan Volkmar Vill Institut fuÈr Organische Chemie, UniversitaÈt Hamburg, D-20146 Hamburg, Germany Slobodan ZÏumer Physics Department, University of Ljubljana, 1000 Ljubljana, Slovenia

1

Introduction Heinz-Siegfried Kitzerow and Christian Bahr

Structures without mirror symmetry are extensively found in nature. The existence or lack of mirror symmetry plays an important role on all length scales of the Universe, from elementary particles to chemical and biological structures and even astronomical systems. The symmetry or asymmetry can have large consequences, for example, from two di¨erent molecular structures (enantiomers), one being the mirror image of the other, one may be a useful drug, the other highly toxic. Moreover, some physical properties of a pure enantiomer may completely deviate from the nonchiral 1:1 mixture of the two opposite enantiomers. This chapter provides a general introduction to the di¨erent aspects of chirality in nature, and particular in liquid crystals.

1.1 1.1.1

General Aspects of Chirality Examples of Chirality in Nature

In 1848, Louis Pasteur [1] noticed that a salt of tartaric acid forms two types of crystals, each one being the mirror image of the other. After separating the two types and preparing two solutions, he found that one solution rotates the plane of linearly polarized light clockwise, the other counterclockwise. This observation of optical activity was a milestone in the investigation of symmetry properties, and in particular of chirality in nature. Chiral objects are objects which cannot be transformed into their mirror image by rotations or translations. Thus, a handedness can be de®ned for chiral objects, for example, a screw is said to be right-handed if its windings propagate clockwise along the screw axis, otherwise it is called left-handed. The name chirality originates from the Greek vocabulary weiÂr (hand), since a hand is a striking example of a chiral object. Other examples are snail's shells or creeping plants. However, the appearance of chirality is not restricted to macroscopic objects. The existence or lack of chirality is also an important feature of molecules and elementary particles, for example, a spinning particle which moves along the spinning axis is chiral (Figure 1.1). 1

2

H.-S. Kitzerow and C. Bahr

Figure 1.1. Chiral objects: Quartz crystals, screws, hands, snail vertigo antivertigo, organic molecules with asymmetric substitution at a carbon atom (amino acids Sand R-alanine), spinning particle. The latter is called right-handed (helicity ˆ ‡1) if the spin is parallel to the velocity, left-handed (helicity ˆ ÿ1) if the spin is antiparallel to the velocity [12].

Theoretically, one might expect that chiral objects of opposite handedness have the same energy and occur in nature with equal probability. However, it is obvious that many chiral structures occur either exclusively with one handedness, or at least more often than the opposite handedness [2]. For example, the shell of the snail vertigo antivertigo is right-handed (Figure 1.1), other snails have left-handed shells, and for very few species the probabilities of ®nding right- and left-handed shells are equal. Among winding plants, the trumpet honeysuckle (Lonicera sempervirens) winds preferably to the left, whereas the bindweed (Convolvulus arvensis) winds to the rightÐlike the majority of helical plants.

1.1.2

Description of Molecular Chirality

On the molecular level, many organic compounds are chiral because their molecules possess a carbon atom with four di¨erent ligands [3]±[5]. These

1. Introduction

3

Figure 1.2. Chirality rule for organic molecules possessing a chiral center (from [3]).

ligands are not in the same plane but their positions form the corners of a tetrahedron, with the central atom being in the center of the tetrahedron (Figure 1.2). The absolute con®guration of such molecules, ``R'' or ``S,'' is described by the chirality rule developed by Cahn, Ingold, and Prelog [6]. According to sequence rules, the four ligands are arranged in a priority list, e.g., A > B > D > E. Ligands of higher atomic number precede ligands of lower atomic number. If the consideration of the nearest neighbors of the chiral center does not result in an unambiguous sequence, the next sphere of ligands is to be investigated until the priority sequence is found. The molecule is then viewed from the side opposite to the ligand with lowest priority, E. If the sense of direction of the three remaining ligands in the sequence A ! B ! D is clockwise, the con®gurational descriptor is R (for Latin rectus, right), and if it is counterclockwise, it is S (for Latin sinister, left). In order to characterize the chirality of a compound, one usually indicates not only the absolute con®guration but also the sign of the optical activity, …‡† or …ÿ†. If the solution of the chiral compound rotates the plane of polarized light clockwise, the sign of its optical activity is positive …‡†. The di¨erent absolute con®guration of two enantiomers corresponds to a di¨erent sign of their optical activities. However, it is not obvious whether the R-con®guration corresponds to positive optical activity …‡† and the S-con®guration to negative optical activity …ÿ†, or vice versa. In connection with amino acids and carbohydrates, the con®guration is usually designated by the pre®x ``D'' or ``L,'' instead of using the absolute con®guration (R or S). If an a-amino acid is drawn in the Fischer projection with the carboxylate group written on top (Figure 1.3), its con®guration is ``L'' (for Latin laevus, left) if the amino (or ammonio) group is on the left, ``D'' (for Latin dexter, right) if this group is on the right side of the Fischer projection formula. For carbohydrates, the highest numbered chiral CHOH group (farthest from the carbonyl end) needs to be considered. If the Fischer projection formula is written with the CbO group on top and the CH2 OH group on the bottom, the con®guration is D if the OH group is on the right, and it is L if the OH group is on the left.

4

H.-S. Kitzerow and C. Bahr

Figure 1.3. Con®guration of a-amino acids and monosaccharides.

1.1.3

Chiral Homogeneity of Biomolecules

The preference of one-handedness appears not only in the shape of shells and plants but also on the biomolecular level [7]. Natural L-amino acids occur much more frequently than D-amino acids. Proteins consist even exclusively of L-amino acids [2], [3]. The occurrence of only one enantiomer is called homochirality. The chirality of the amino acids is re¯ected in the secondary structure of polypeptides, the a-helix (Figure 1.4(a)) [8], which is formed due to hydrogen bonding. In natural polypeptides, the a-helix is always righthanded. Some proteins serve as enzymes, i.e., biocatalysts. Due to the chirality, their activity can be very sensitive to the absolute con®guration of the substrate molecules. Other proteins which contain many prolin units form collagen ®bers which are very stable due to a tripel helical structure (Figure 1.4(b)). The pitch of the tripel helix (0.96 nm) is larger than the pitch of the

Figure 1.4. Chiral supramolecular structures: (a) a-helix of polypeptides, (b) polyprolin-helix of collagene, and (c) deoxyribonucleic acid (DNA): Z ˆ D-Deoxyribose, P ˆ phosphoric acid, A ˆ adenine, G ˆ guanine, C ˆ cytosine, T ˆ thymine (from [8]).

1. Introduction

5

a-helix (0.54 nm) and the helix forms very rigid ®bers with high tensile strength (with a length of 300 nm and a diameter of 1.5 nm) which are responsible for the strength of the connective tissue. In addition to the preference of L-amino acids in biochemical processes, there is a preference for D-sugars. Again, the preference of one enantiomer for the monomer units leads to a helical superstructure of corresponding macromolecules, for example, the desoxyribonucleic acid (DNA) consists of a backbone formed by esters of phosphoric acid with the monosaccharide D-desoxyribose. The base pairs adenine, cytosine, guanine, and thymine are attached to this backbone. Two polymer chains connected by hydrogen bonds form the DNA double helix which is right-handed (Figure 1.4(c)). (The function of a rather rare left-handed form (Z-DNA) is still under discussion [8].) Due to the chirality of receptor molecules, the taste, odor, or drug activity of enantiomers can be very di¨erent (Table 1.1) [3]±[5]. For example, (R)-…‡†-limonene smells like oranges whereas (S)-…ÿ†-limonene smells like lemons, (R)-…ÿ†-carvone smells like spearmint, whereas (S)-…‡†-carvone has the odor perception of caraway. One of the compounds to which our sense of taste is most sensitive, is (R)-…‡†-1-p-menthene-8-thiol which has the ¯avor of grapefruit and could be perceived at a concentration of 0.02 ng/l. Its enantiomer has the same ¯avor but the concentration needs to be four times higher in order to be noticed [9]. In order to highlight the di¨erent biological activity of isomers, the more active isomer is often called eutomer, whereas the less potent or inactive isomer is called distomer. A disastrous example of the di¨erent activities of the eutomer and the distomer was the use of contergan as an antineusea agent for pregnant women in the early 1960s. The distributed medicament contained the racemic mixture. While one enantiomer had the intended e¨ect, the other turned out to cause serious birth defects of the babies. Since then, the relation between the chirality and the physiological e¨ect of drugs is very attentively controlled.

1.1.4

Parity Violation of the Weak Interaction

The naive assumption that left- and right-handed structures are energetically equivalent is not valid on the subnuclear length scale. The observation by Wu et al. [10] that the radioactive b-decay of 60 Co generates more lefthanded electrons than right-handed electrons con®rmed the proposal [11] that parity is not conserved in the weak interactions. While three fundamental forcesÐgravity, the eletromagnetic force, and the strong forceÐare independent of handedness, it became apparent that the fourth force, the weak force is chirally asymmetric. Later the neutrino was found to be always left-handed, whereas the antineutrino is always right-handed [12]. One of the fascinating mysteries of science is the question whether the parity violation of the weak force and the homochirality of biomolecules are related to each other [7]. According to the uni®ed theory of electroweak interactions (Standard Model) by Glashow, Salam, and Weinberg [13], the weak and the

6

H.-S. Kitzerow and C. Bahr

Table 1.1. Enantiomers with stereoselective properties. The data are from [3] and [5], where more examples can be found.

electromagnetic force are di¨erent aspects of a single electroweak interaction. The weak interaction between particles is associated with weak charged currents (W force) or weak neutral currents (Z force). The corresponding weak W charge is nonzero for a left-handed electron but zero for a righthanded electron. This asymmetry leads to the preferred production of left-

1. Introduction

7

handed electrons in the nuclear b-decay. In spite of the very short range of the weak interaction, it causes a chirality of atoms. The weak Z charge has opposite sign for left- and right-handed electrons. Due to the corresponding Z force, the electrons of an atom are either attracted to the nucleus or repelled, depending on their handedness. Due to the Z force, the paths of electrons in an atom become chiral. This chirality of atoms was con®rmed by optical experiments [14] which demonstrate that the mirror symmetry in stable atoms is broken during absorption of light. This symmetry breaking has consequences on the molecular level. The spin±orbit coupling due to electromagnetic interactions tends to align the axis of each electron's orbit against its axis of spin. In a molecule with helical shape, the spin±orbit coupling causes a preference of electrons with one-handedness [2]. Consequently, two enantiomers have slightly di¨erent energies due to the Z force. According to the concept of ``true'' and ``false'' enantiomers by Barron [15], chemical enantiomers made of matter have to be considered as false enantiomers. Two true enantiomers (which are energetically equivalent) would need to consist of matter and antimatter, respectively, so that not only the absolute molecular con®gurations but also the weak Z charges are opposite. A possible link between the asymmetry of the b-decay and the chirality of biomolecules was proposed by Ulbricht and Vester [16]. Since the emitted electrons cause circularly polarized g radiation (Bremsstrahlung), it was proposed that stereoselective photochemical reactions may have caused the accumulation of certain enantiomers in nature. Although tests of this hypothesis lead to contradicting results, it is known today that circularly polarized light can be used for either asymmetric synthesis (e.g., the synthesis of hexahelicene) or stereoselective photodestruction of racemic mixtures [17]. Another possible explanation for the chiral homogeneity in nature is the di¨erent stability of enantiomers due to the weak interaction. Quantum mechanical calculations [7], [18] show that the L-enantiomers of naturally occurring amino acids and the D-enantiomers of sugars have indeed a lower energy than their unpreferred enantiomers (DE A ÿ10ÿ14


ÿ10ÿ13 J/mol [7]). In order to explain the optical activity of living matter, Frank [19] has proposed a kinetic model which describes the accumulation of one enantiomer due to spontaneous symmetry breaking and autocatalytic enhancement of its concentration. Although the energy di¨erence between D- and L-enantiomers is extremely small, Frank's model has been extended in order to discuss whether a very small systematic chiral perturbation rather than an accidental event can determine which enantiomer is formed [20]±[23]. For example, Kondepudi and Nelson [23] considered the reaction scheme S ‡ T T L…D†; S ‡ T ‡ L…D† T 2L…D†; L ‡ D ! P;

…1:1† …1:2† …1:3†

8

H.-S. Kitzerow and C. Bahr

Figure 1.5. (a) Bifurcation in the autocatalytic nonequilibrium system described by (1.1)±(1.3) may lead to spontaneous symmetry braking. (b) Stable states and calculated trajectory of the same systems under the in¯uence of random ¯uctuations and a weak, but systematic external chiral in¯uence (from [23]).

where the chiral enantiomers L or D are formed (1) from nonchiral substrates either directly, or (2) through an autocatalytical reaction, and (3) irreversibly removed by reaction. The system is driven far from equilibrium due to a suitable supply of S and T, and the chiral dissymmetry a :ˆ 1 2([L] ÿ [D]) is studied as a function of the parameter l :ˆ ‰SŠ‰TŠ (where ‰ Š denote concentrations). Without ¯uctuations, this system shows a critical point and a bifurcation (Figure 1.5(a)). However, if both random ¯uctuations and an external chiral in¯uence (such as that from circularly polarized light) are taken into account (Figure 1.5(b)), the solutions of the stochastic equations show a 98% chance that the enantiomer favored by the weak neutral current will emerge dominant even if the rms values of random chiral in¯uences are ®ve orders of magnitude larger than the weak neutral current e¨ect. It was concluded that a systematic chiral in¯uence over a period of A15,000 years is su½cient to determine which enantiomer will dominate [23]. Although it is tempting to believe that the development of the enantiomeric homogeneity in nature was caused by the asymmetry of the weak interaction, an experimental proof of this assumption is beyond human capabilities.

1.1.5

Point Group Symmetry and Variety of Chiral Compounds

The most e½cient way to characterize the symmetry of a molecule is the determination of its symmetry elements. Each symmetry operation (inversion, rotation, re¯ection, or a combination of the latter two) which leaves the molecule unchanged, de®nes an inversion point (i), an n-fold rotational axis …Cn † [corresponding to a rotation by 2p=n], a mirror plane …s†, or a rotation± re¯ection axis …Sn †. The set of symmetry elements is the point group. It

1. Introduction

9

may be designated in the SchoÈn¯ies (or Herrmann±Mauguin) notation. An object is chiral if it cannot be projected on itself by any combination of a rotation and a re¯ection. Its point group must not contain any rotation± re¯ection axis Sn . Thus, all objects which exhibit a mirror plane …s ˆ S1 † are achiral (ˆ nonchiral). However, the lack of a mirror plane is not su½cient to prove that an object is chiral. All objects which show inversion symmetry …i ˆ S2 ˆ sh ‡ C2 † or a higher rotation±re¯ection axis Sn are achiral, too. Thus, only the point groups Cn (1, 2, 3, . . . ), Dn (222, 32, 422, 622, . . . ), T(23), O(432), and I correspond to chiral objects. However, achiral objects are described by one of the point groups Ci (1), Cs (m), Cnv (2mm, 3m, 4mm, 6mm, . . . ), Cnh (2=m, 6, 4=m, 6=m, . . . ), Dnh (mmm, 62m,4=mmm, 6=mmm, . . . ), Dnd (42m,3m,4,3, . . . ), Td (43m), Th (m3), Oh (m3m), or Ih [24]. Organic compounds which possess a chiral center, i.e., a carbon atom with four di¨erent ligands, have the lowest possible symmetry [point group C1 …1†]. Two isomers with di¨erent absolute con®gurations, R or S, can occur for a molecule with one chiral center. The pure chiral compound with Rcon®guration is the enantiomer of the pure compound with S-con®guration, and vice versa. Enantiomers possess di¨erent signs of the optical activity, but their phase transition temperatures and other physical properties are equal. However, the 1:1 mixture of two enantiomers, the racemic mixture, is nonchiral. Its optical activity is zero, and its transition temperatures can deviate from the pure chiral compounds. Molecules with two or more asymmetrically substituted atoms possess not only an enantiomer (its mirror image), but also diastereomers, isomers which are not related as a mirror image. Diastereomers exhibit di¨erent physical properties, e.g., …R; R†-sodium rubidium tartrate (Figure 1.6) is the enantiomer of …S; S†-sodium rubidium tartrate, and …R; S†-sodium rubidium tartrate is the enantiomer of …S; R†-sodium rubidium tartrate. The molecule with …R; R†-con®guration is the mirror image of …S; S†-con®guration, and consequently the chiral …R; R†-compound shows the same melting point as the …S; S†-compound. However, the …R; S†-molecule shows a di¨erent molecular symmetry than …R; R† and …S; S†, and thus a di¨erent melting point. …R; S† and …S; R† are enantiomers of each other, and diastereomers of …R; R† and …S; S†. In some cases, a molecule with two asymmetrically substituted carbon atoms may possess mirror symmetry, thus being achiral. This is the case for …R; S†- and …S; R†-tartaric acid where the ligands at the two central carbon atoms are the same. The hypothetical …R; S†- and …S; R†-forms are in fact identical. The corresponding achiral compound is called the mesoform. However, …R; R†-tartaric acid and …S; S†-tartaric acid are chiral and enantiomers of each other. The racemic mixture of …R; R†- and …S; S†-tartaric acid is achiral, but its physical properties di¨er from the mesoform. The example of tartaric acid shows that molecules with asymmetrically substituted carbon atoms may be achiral. On the other hand, many chiral organic compounds contain no asymmetrically substituted carbon atom. Typical examples are axially twisted molecules like 1,3-dichlorallen, twisted

10

H.-S. Kitzerow and C. Bahr

Figure 1.6. Stereoisomers of sodium rubidium tartrate. …R; R† is the enantiomer of …S; S†. Top: Newman projection; Bottom: Fischer projection. …R; S† is the enantiomer of …S; R†. Both …R; R† and …S; S† are diastereomers of …R; S† and …S; R†. For disodium tartrate or for tartaric acid, there exists only one nonchiral mesoform instead of the enantiomers …R; S† and …S; R†.

biphenyles which contain the same substituents at the two rings, or helicenes (Figure 1.7). Molecules with axial chirality are described by the point group C2 or exhibit even higher symmetry (Cn or Dn ). The absolute con®guration of helicenes is indicated by the pre®x P- or M- [3]. An extreme example of an axially chiral molecule is the twisted aromatic molecule shown in Figure 1.7(d), which exhibits an intramolecular end-to-end twist of 105 [25]. Figure 1.7(e) shows a paracyclophane molecule which exhibits a chiral plane. Here, the chirality is due to the fact that three di¨erently substituted neighboring atoms …a; b; c† are connected to a fourth neighbor (marked by the arrow) which is not in the plane de®ned by a, b, and c. In this case, the plane is viewed from the out-of-plane atom closest to the chiral plane. Then, the descriptor is pR if the atoms a, b, c describe a clockwise array, otherwise pS [3].

1.1.6

Stereoselective Synthesis

The very important role of chirality has motivated the development of very sophisticated methods for the synthesis of optically active compounds. Three di¨erent strategies can be used in order to obtain a pure optically active compound which contains only one of two enantiomers. One can either: (1) start from a pure optically active compound making use of the natural pool of chiral compounds and try to use only chemical reactions in which the absolute con®guration at the chiral center is either maintained or completely reversed; or

1. Introduction

11

Figure 1.7. Axially chiral molecules (a)±(d) and a molecule with a chiral plane (e). (a) Dichlorallen, (b) twisted biphenyl, (c) helicene, (d) highly twisted aromatic compound [25], and (e) paracyclophane.

(2) use chiral catalysts, and in particular enzymes, in order to synthesize a chiral compound starting from nonchiral reactands; or (3) synthesize a racemic mixture and try to separate the two enantiomers of the product. The chiral pool refers to readily available optically active natural products, some of which are commercially used in quantities of 10 2 ±10 5 tonnes per year [5]. Among them, the most inexpensive compounds are a-amino acids, like monosodium L-glutamate, or carbohydrates, like dextrose or sorbitol. The success of the second method depends on the availability of particular catalysts. One rather special example, how e½cient stereoselective synthesis can work, is the syn-selective aldol reaction followed by a stereoselective alkene hydroboration and ketone reduction (Figure 1.8) which were used by Paterson et al. [26] to synthesize intermediates for the antibiotic oleandomycin. According to Paterson et al., four new stereocenters are formed in only two synthetic steps [9]. For the purpose of separating racemic mixtures

12

H.-S. Kitzerow and C. Bahr

Figure 1.8. Stereoselective synthesis according to Paterson et al. [26].

(case 3), a chemical method is the reaction with a pure chiral compound in order to form diastereomers which can then be separated by recrystallization or similar methods. An attractive method for racemic mixtures forming conglomerates (as opposed to racemic compounds) is the direct crystallization of one enantiomer by seeding a supersaturated racemic mixture with appropriate chiral crystallites. In addition, chiral stationary phases for chromatography are available, and their development is in further progress. Today, the chiral compound with the largest number of stereocenters which was synthesized in a chemical laboratory is palytoxin (Figure 1.9), synthesized by Suh and Kishi [27]. This compound shows 64 stereocenters with a de®ned absolute con®guration. The number of its stereoisomers is 2 64 A 1:8  10 19 which demonstrates the di½culties of this synthesis.

Figure 1.9. Structure of palytoxin [27].

1. Introduction

13

Figure 1.10. Typical phase sequence in thermotropic liquid crystals. Top: Calamitic liquid crystals (consisting of rod-like molecules); Bottom: Discotic liquid crystals (consisting of disk-shaped molecules).

1.2

Impacts of Chirality in Liquid Crystals

Chirality is a very important parameter to be considered in liquid crystals [28]±[33]. The liquid crystalline state is a state of matter which is characterized by the alignment of molecules or molecular aggregates. Thermotropic liquid crystals consist of rod-like or disk-like molecules, and show one or several additional phase(s) in the temperature range between the crystalline and the isotropic liquid state (Figure 1.10). These mesophases are ¯uid, but anisotropic. Their physical properties depend on the orientation, like the properties of a crystal. The structure of liquid crystalline phases is characterized by a long-range orientational order of the molecules. However, the centers of the molecules are free to move and show either no positional order (nematic phases) or a quasi-long range positional order in less than three dimensions (smectic and columnar phases). An overview about the classi®cation of smectic phases is given in Figure 1.11 (for more details, see Chapter 8). Chiral columnar liquid crystals are extensively described in Chapter 11. A di¨erent kind of mesophase is formed by amphiphilic molecules in lyotropic liquid crystals. Amphiphilic molecules exhibit a hydrophilic (polar) head and a hydrophobic (nonpolar) tail and form micelles, columns, or lamellae consisting of many molecules. These units can be arranged in an anisotropic structure, if the concentration of the amphiphilic substance in a solvent is suitable. In contrast to thermotropic liquid crystals, these anisotropic solutions of amphiphilic molecules are called lyotropic liquid crystals. In this book, we consider mainly thermotropic liquid crystals; an overview on chiral lyotropic liquid crystals is given in Chapter 14. The characteristic feature of thermotropic liquid crystals is the preferred orientation of molecules along a certain direction which is described by a

14

H.-S. Kitzerow and C. Bahr

Figure 1.11. Structures of various smectic liquid crystals. The side view of the respective layer structure is shown for each phase to the left. The molecules are represented by ellipses. The plan view of the layers for the respective phase is shown to the right. The cross-sections of the molecules are represented by circles if the molecules are not tilted. Triangles indicate the tilt direction. The dotted lines are the lattice sides for long-range order. The smectic B, I, and F phases have bond orientational order in that the hexagonal packing net has the same long-range orientation, even though the positional order is short range. Courtesy by J.W. Goodby (from [31].)

unit vector, the director n. The local direction of n can continuously vary in space, i.e., n ˆ n(r). It is important to note that the parallel alignment in nematic liquid crystals is not connected with polar ordering, i.e., n and ÿn are equivalent. For many problems, e.g., the numerical calculation of defects in the director ®eld or the consideration of biaxial structures, it is necessary to use the more appropriate description by an alignment tensor. Its components aij are related to the director components ni by aij ˆ ni nj ÿ 13 dij :

…1:4†

However, it is popular to use the director as far as possible to visualize liquid crystal structures. The director ®eld shows elastic behavior, and can be in¯uenced by external electric and magnetic ®elds. At interfaces, the orientation of n is usually anchored due to interactions between the liquid crystal

1. Introduction

15

molecules and the substrate. An alignment of the molecules parallel to the substrate (homogeneous anchoring) or perpendicular to the substrate (homeotropic anchoring) can be achieved by suitable surface treatment or coating of the substrates [34]. The most obvious anisotropic property of liquid crystals is their birefringence. A nonchiral nematic liquid crystal is optically uniaxial, and the optical axis is parallel to the director. For rodlike molecules, the polarizability along the molecular axis is larger than the polarizability perpendicular to the molecular axis, and consequently the extraordinary refractive index ne (which is e¨ective for linearly polarized light with the plane of polarization parallel to the director) is larger than the ordinary refractive index no (which is e¨ective if the plane of polarization is perpendicular to the director).

1.2.1

Electro-optic Application

The unique combination of high birefringence (typically ne ÿ no A 0:2) with the possibility to modulate either the direction of the optical axis or the optical retardation by an external electric ®eld is the basis for the electrooptic application of liquid crystals. In many of these applications, the birefringent optical retardation by a particularly aligned liquid crystal is used. If the liquid crystal cell is placed between two linear polarizers, the state of polarization of the light propagating through the liquid crystal is changed. Any variation of the polarization of the light trasmitted through the liquid crystal results in a change of the light intensity passing the second polarizer. In twisted nematic (TN) liquid crystal displays [35], the director shows a parallel anchoring on both glass substrates, but the alignment directions at the two substrates are twisted by 90 with respect to each other (Figure 1.12). As a consequence, the director shows a continuously twisted structure in the ®eld-o¨ state. Due to a waveguiding e¨ect, the plane of polarized light propagating through the sample is rotated by 90 , and the light can pass the second polarizer. The application of an electric ®eld perpendicular to the substrate causes an alignment of the director along the ®eld direction. In this state, the liquid crystal does not change the polarization of the transmitted light and the linearly polarized light passing the ®rst polarizer is blocked by the second polarizer. Apart from various improvements of the switching characteristics [36], the angular dependence of the contrast can be considerably improved by using the in-plane switching (IPS) geometry [37] where a liquid crystal with negative dielectric anisotropy is used and the ®eld is applied in the cell plane. Note that the TN setup is a chiral structure. In principle, it is possible to build a TN±LCD with a nonchiral nematic liquid crystal because the 90 twist can be induced by appropriate surface preparation. However, for practical applications chiral additives are used in order to stabilize one-handedness of the twisted structure, thereby avoiding the occurrence of domains with opposite handedness which would be separated by unwanted domain walls.

16

H.-S. Kitzerow and C. Bahr

Figure 1.12. Twisted nematic (TN) liquid crystal display (LCD). Courtesy by M. Schadt (from [36]).

1.2.2

Helical Structures

The presence of chiral molecules in a liquid crystal can have various consequences. Chirality may cause an intrinsic helical structure of the director ®eld (Figure 1.13). Instead of the uniform alignment of the director ®eld occurring in the nematic phase, the respective chiral nematic phase exhibits a helical structure (Figure 1.13(a)). This structure shows a uniform helix axis. The director is perpendicular to this axis and its azimuthal angle changes continuously. Chiral nematic phases were ®rst observed in cholesteryl derivatives. Thus, the chiral nematic phase is also called a cholesteric phase. In spite of this historical name, the structure of cholesteric molecules can considerably vary from cholesterol. The cholesteric phase is completely miscible with the nonchiral nematic phase occurring in the respective racemic mixture. In ideal chiral±racemic mixtures, the reciprocal value of the pitch p is proportional to the enantiomeric excess. A very striking phenomenon is the appearance of Bragg re¯ection if the pitch of the helical structure is of the same order as the wavelength of visible light [38]. This selective re¯ection of circularly polarized light shows a brillant color e¨ect with a very characteristic angular dependence of the color. The impressive color e¨ect was prob-

1. Introduction

17

Figure 1.13. Helical structures formed in chiral liquid crystals. (a) Cholesteric phase (N*), (b) blue phase (BP), (c) chiral smectic-C phase (SmC*), and (d) twist grain boundary phase (TGB).

ably responsible for the ®rst observations of liquid crystals [39], [40]. It is not restricted to calamitic compounds, but occurs also in cholesteric discotic liquid crystals [41]. Selective re¯ection is used for the application of liquid crystals in thermometers (making use of the temperature dependence of the Bragg wavelength) [42], [43], polarizing mirrors [44], decorative objects (paintings, mouse pads, business cards), re¯ective electro-optic displays [45]± [47], and optical storage [48]±[50]. In the last few years, even color pigments for car paintings are being developed in order to make use of the unique color e¨ects [51], [52]. Apart from selective re¯ection, the ¯exoelectric behavior [53], the cholesteric±nematic phase change e¨ect [54], [55], and scattering e¨ects in cholesteric gels [56], [57] also provide interesting electro-optic responses. In more detail, the basic optical properties of cholesteric liquid crystals are described in Chapter 2 (and in Annex I and II), the induction of helical structures by chiral dopants in a nematic host is discussed in Chapter 3, and special electro-optical and nonlinear optical e¨ects are addressed in Chapter 6. Blue phases (Chapter 7) and cholesteric liquid crystals in certain con®ned geometries can show double twist (Figure 1.13(b)), a twisted structure which cannot be described by a single twist axis. For the theoretical analysis of

18

H.-S. Kitzerow and C. Bahr

such structures, it is instructive to investigate a twist tensor Tij [58] which is related to the director components ni by Tij :ˆ …eikl nk nl; j †=…nm nm †:

…1:5†

The number of its nonzero eigenvalues (0, 1, or 2) indicates the number of twist axes. The latter are given by the eigenvectors. The corresponding eigenvalues indicate the local twist 2p=p [58]. A helical director ®eld also occurs in the chiral smectic-C phase and those smectic phases where the director is tilted with respect to the layer normal (Figure 1.13(c)). In these cases, the pitch axis is parallel to the layer normal and the director inclined with respect to the pitch axis. Very complicated defect structures can occur in the temperature range between the cholesteric (or isotropic) phase and a smectic phase. The incompatibility between a cholesteric-like helical director ®eld (with the director perpendicular to the pitch axis) and a smectic layer structure (with the layer normal parallel or almost parallel to the director) leads to the appearance of grain boundaries which in turn consist of a regular lattice of screw dislocations. The resulting structures of twist grain boundary phases are currently extensively studied. The state of the art in this topical ®eld is summarized in Chapter 10.

1.2.3

Polar Properties

The second fundamental type of consequence of chirality (apart from inducing a helical director ®eld) is the appearance of polar physical properties which are described by a vector or a third-rank tensor. A polar property which is described by a vector can never occur perpendicular to a rotational axis or mirror plane of the considered structure. A very important property of this type is the spontaneous polarization Ps which gives rise to the ferroelectric properties of certain liquid crystals [59] and their electro-optic applications [60], [61]. For nonchiral liquid crystals, a spontaneous polarization can neither occur in the nematic phase (point group DLh ), nor in the smectic-A phase (point group DLh ), nor in the smectic-C phase (point group C2 h ). However, the presence of chiral molecules reduces the local symmetry of the respective phases (even when the helical director ®eld is unwound) to the space groups DL (chiral nematic and chiral smectic-A phase) and C2 (chiral smectic-C phase) because of the lack of mirror symmetry Figure 1.14. The latter structure shows only one twofold axis which is perpendicular to the plane de®ned by the director n and the layer normal ql . Thus, a spontaneous polarization in the …ql ; n† plane cannot occur, but a spontaneous polarization along ql  n is possible in chiral smectic-C phases. Similar symmetry arguments can be applied for chiral tilted columnar phases [62]. The ferroelectric and antiferroelectric properties of chiral smectic liquid crystals are very promising for electro-optic applications. They show bi- and multistable switching behavior and rather short switching times (down to 1 ms). In surface stabilized ferroelectric liquid crystal displays [63], a uniformly parallel

1. Introduction

19

Figure 1.14. Symmetry elements of the nonchiral smectic-C (SmC) phase (point group C2 h ), and the chiral smectic-C (SmC*) phase (point group C2 ). For both phases, the rotation around the C2 -axis by p transforms a hypothetical polarization P ˆ …Px ; Py ; Pz † into …ÿPx ; Py ; ÿPz †. Therefore, Px and Pz must be zero due to the symmetry of these phases. The additional symmetry element sh in the SmC phase transforms …Px ; Py ; Pz † into …Px ; ÿPy ; Pz †, so that Py ˆ 0 in SmC. However, the same arrangement of chiral molecules shows no mirror symmetry. Thus, Py 0 0 is allowed in the SmC* phase.

aligned smectic liquid crystal is used which serves as an optical waveplate. Two parallel polarizers are used in this case. Since the tilt angle between the director n and the smectic layer normal ql is coupled to the modulus of the electric polarization …Ps z ql  n†, the optical axis of the liquid crystalline waveplate can be rotated within the plane perpendicular to the ®eld direction. In a setup of this kind, the director ®eld shows no helical structure, but the presence of chiral molecules is a precondition for a nonzero spontaneous polarization. Other applications make use of the electroclinic e¨ect [64] (here again chiral molecules need to be present for a nonzero electroclinic coe½cient), the helical unwinding of the smectic-C* helix [65], or the con®nement of a ferroelectric liquid crystal in cells similar to the TN cell [66]. The advantages and peculiarities of ferroelectric and antiferroelectric liquid crystals are described in Chapters 8 and 9.

20

H.-S. Kitzerow and C. Bahr

Chirality can also lead to certain nonlinear optical e¨ects. In general, the dipole moment induced in a molecule by an electric ®eld E has the components mi ˆ aij Ej ‡ bijk Ej Ek ‡ gijkl Ej Ek El ‡


;

…1:6†

where a is the molecular polarizability and b; g; . . . are hyperpolarizabilities. The molecular hyperpolarizability b is large for molecules showing a large conjugated p-electron system with asymmetrically attached electron donor and acceptor groups. The corresponding macroscopic electric polarization is given by …1†

…2†

…3†

Pi ˆ wij Ej ‡ wijk Ej Ek ‡ wijkl Ej Ek El ‡


;

…1:7†

where w…1† is the linear dielectric susceptibility and w…2† , w…3† ; . . . are nonlinear susceptibilities. The second-order nonlinear susceptibility w…2† , a third rank tensor, vanishes in centrosymmetric structures. Thus, a noncentrosymmetric arrangement is necessary for all nonlinear e¨ects which are connected to w…2† , e.g., the second harmonic generation. This condition may be ful®lled due to the vicinity of a surface (which brakes mirror symmetry), due to poling by an external ®eld, or due to the spontaneous appearance of a chiral structure, e.g., built by chiral molecules. As a consequence, relatively large nonlinear susceptibilities w…2† have been observed in chiral smectic-C phases [67]±[71]. Some examples of w…2† -values found in chiral smectic-C phases are given in Table 1.2 [67]±[72]. Extended reviews about nonlinear optical e¨ects were given by Khoo and Wu [73] and Pal¨y-Muhoray [74]. Some of the interesting nonlinear optical e¨ects occurring in cholesteric phases are described in Chapter 6 of this book.

1.2.4

Unique Phases

It is important to note that also nonchiral molecules are capable of forming chiral mesophases. In particular, molecules with a bent core (``bananashaped'' molecules) can build polar, and even chiral liquid crystal structures [75]±[78]. Bent-core molecules form a variety of new phases (B1±B7, Table 1.3) which di¨er from the usual smectic and columnar phases (see also Chapter 8). As a consequence of the polar arrangement, antiferroelectric-like switching was observed in the B2 phase formed by bent-core molecules, and second harmonic generation was found in both the B2 phase and the B4 phase. The latter phase is probably a solid crystal. It consists of two domains showing selective re¯ection with opposite handedness. In the liquid crystal…2† line B2 phase, the e¨ective nonlinear susceptibility weff can be modulated by an external dc ®eld [79] (Figure 1.15). In conclusion, chirality plays an important role for many applications of liquid crystals, in particular for electro-optic displays and thermochromic temperature displays. In addition, chiral liquid crystals show interesting helical structures, defect structures, and even additional phases which do not

1. Introduction

21

Table 1.2. Nonlinear susceptibilities w…2† observed in ferroelectric liquid crystals and in the phases B2 and B4 formed by bent-core molecules. The indices are given in the following coordinates: y-axiskspontaneous polarization; z-axisksmectic layer normal; x-axiskprojection of the director in the plane of the smectic layers. For comparison, the values of two NLO crystals are given.

occur in non-chiral liquid crystals, e.g., blue phases and TGB phases. As an example, the appearance of the smectic-Q phase in a chiral±racemic phase diagram [80] is represented in Figure 1.16. The occurrence of certain polar properties requires the breaking of mirror symmetry, and thus only chiral liquid crystals are capable of showing ferroelectric, antiferroelectric, or electroclinic switching behavior, or nonlinear properties which are connected with the second-order nonlinear susceptibility. The following chapters pro-

22

Table 1.3. Overview of mesophases (B1±B7) which were observed so far for molecules with a bent core (``banana shape''). Courtesy by S. Diele (from [78]).

H.-S. Kitzerow and C. Bahr

1. Introduction

23

Figure 1.15. Intensity of a second harmonic generation as a function of applied electric ®eld strength in the phase B2 which is formed by achiral bent-core molecules. Courtesy by F. Kentischer and R. Macdonald (from [79]).

vide a consideration about the formation of chiral structures by chiral molecules (Chapter 3), a brief survey of the chemical structures of chiral liquid crystals (Chapter 4), and reviews about some of the remarkable properties of chiral liquid crystals.

Figure 1.16. Chiral±racemic phase diagram showing the appearance of a smectic phase (SmQ) in chiral mixtures with su½cient enantiomeric excess. The molecule of the investigated compound (M7BBM7) shows two chiral centers. The concentration 0% corresponds to the racemic mixture of the …R; R† and …S; S† isomers, 100% corresponds to the pure …S; S† enantiomer. Courtesy by D. Bennemann (from [80]).

24

1.2.5

H.-S. Kitzerow and C. Bahr

Life

Last but not least, it is remarkable that mesophases and living systems are closely connected. One aspect of this relation is the formation of bilayers, micelles, and other organized structures by amphiphilic molecules. These supermolecular structures can form lyotropic mesophases (Chapter 14) and vesicles [81]. In addition, Lehn and coworkers [82] have demonstrated a selfassembly of supramolecular liquid crystalline polymers (Figure 1.17) which is very reminiscent of the supermolecular structure of biochemical macromolecules (Figure 1.5). Tartaric acid-based molecules bearing complemen-

Figure 1.17. Supramolecular structure of a liquid crystalline polymer. Top: Molecular structure. Lower left: Triple helical superstructure derived from X-ray data. Lower right: Helical textures observed by electron microscopy: (A) L-enantiomers, (B) D-enantiomers, and (C) mesoform. Courtesy by J.-M. Lehn (from [82]).

1. Introduction

25

tary hydrogen bonding units (2,6-diaminopyridine and uracil units) form a triple-helical superstructure. These systems show a remarkable example of chiral recognition. A mixture of L and D compounds demixes spontaneously, thereby forming left- and right-handed helices. However, no helicity is observed for the mesocompounds. Patricia Cladis illustrates a further aspect of the relation between chirality and living systems (Chapter 15). The appearance of a characteristic length (namely the pitch) in helical structures is a precondition for the pattern formation in certain nonequilibrium systems made of liquid crystals.

References [1] L. Pasteur, C. R. Hebd. SeÂan. Acad. Sci. Paris 26, 535 (1848). [2] R.A. Hegstrom and D.K. Kondepudi, Scienti®c American, January 1990, 98 (1990). [3] E.L. Eliel, S.H. Wilen, and L.N. Mander, Stereochemistry of Organic Compounds, Wiley, New York, 1994. [4] D. Nasipuri, Stereochemistry of Organic Compounds: Principles and Applications, Wiley Eastern, New Delhi, 1991. [5] A.N. Collins, G.N. Sheldrake, and J. Crosby (Eds.), Chirality in Industry: The Commercial Manufacture and Applications of Optically Active Compounds, Wiley, Chichester, 1992. [6] R.S. Cahn, C.K. Ingold, and V. Prelog, Angew. Chem. Int. Ed. Engl. 5, 385 (1966). [7] R. Janoschek (Ed.), ChiralityÐFrom Weak Bosons to the a-Helix, SpringerVerlag, Berlin, 1991. [8] Lexikon der Biologie Spektrum Akademischer, Verlag, Heidelberg, Berlin, 1996. [9] R. Faust, G. Knaus, U. Siemeling, and H.-J. Quadbeck-Seeger, ChemieRekorde: Menschen, MaÈrkte, MolekuÈle, Wiley-VCH, Weinheim, 1997. [10] C.S. Wu, E. Ambler, R.W. Hayward, D.D. Hoppes, and R.P. Hudson, Phys. Rev. 105, 1413 (1957). [11] T.D. Lee and C.N. Yang, Phys. Rev. 104, 254 (1956). [12] see, e.g., D. Gri½ths, Introduction to Elementary Particles, Harper & Row, New York, 1987. [13] S. Weinberg, Rev. Mod. Phys. 52, 515 (1980); A. Salam, Rev. Mod. Phys. 52, 525 (1980). [14] M.-A. Bouchiat and L. Pottier, Science 234, 1203 (1986). [15] L.D. Barron, J. Am. Chem. Soc. 108, 5539 (1986). [16] T.L.V. Ulbricht and F. Vester, Tetrahedron 18, 629 (1962). [17] W.A. Bonner, Top. Stereochem. 18, 1 (1988). [18] S.F. Mason and G.E. Tranter, J. Chem. Soc., Chem. Commun. 117 (1983). [19] F.C. Frank, Biochimica Biophysica Acta 11, 459 (1953). [20] K. Tennakone, Chem. Phys. Lett. 105, 444 (1984). [21] S.F. Mason, Nature 311, 19 (1984). [22] S.F. Mason, Nature 314, 400 (1985). [23] D.K. Kondepudi and G.W. Nelson, Nature 314, 438 (1985). [24] P.W. Atkins, Physical Chemistry, Oxford University Press, Oxford, 1994. [25] X. Qiao, D.M. Ho, and R.A. Pascal, Angew. Chem. 109, 1588 (1997).

26

H.-S. Kitzerow and C. Bahr

[26] I. Paterson, R.D. Norcross, R.A. Ward, P. Romea, and M.A. Lister, J. Am. Chem. Soc. 116, 11287 (1994). [27] E.M. Suh and Y. Kishi, J. Am. Chem. Soc. 116, 11205 (1994). [28] P.G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed., Oxford University Press, Oxford, 1993. [29] S. Chandrasekhar, Liquid Crystals, 2nd ed., Cambridge University Press, Cambridge, 1992. [30] D. Demus, J. Goodby, G.W. Gray, H.-W. Spiess, and V. Vill (Eds.), Handbook of Liquid Crystals, Vols. 1±3, Wiley-VCH, Weinheim, 1998. [31] S. Elston and R. Sambles (Eds.), The Optics of Thermotropic Liquid Crystals, Taylor & Francis, London, 1998. [32] J.W. Goodby, J. Mater. Chem. 1, 307±318 (1991). [33] J.W. Goodby, A.J. Slaney, C.J. Booth, I. Nishiyama, J.D. Vuijk, P. Styring, and K.J. Toyne, Mol. Cryst. Liq. Cryst. 243, 231±298 (1994). [34] T. Uchida and H. Seki, Surface alignment in liquid crystals, in: Liquid Crystals. Applications and Uses, (edited by B. Bahadur), Vol. 3, Chap. 5, World Scienti®c, Singapore, 1992. [35] M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127 (1971). [36] M. Schadt, Mol. Cryst. Liq. Cryst. 165, 405 (1988). [37] G. Baur, R. Kiefer, H. Klausmann, and F. Windscheid, Liquid Crystals Today 5, 13 (1995). [38] H. de Vries, Acta Crystallogr. 4, 219 (1951). [39] P. Planer, Liebigs Ann. Chem. 118, 25 (1861). [40] F. Reinitzer, Monatshefte Chemie 9, 421 (1888). [41] D. KruÈerke, H.-S. Kitzerow, G. Heppke, and V. Vill, Ber. Bunsenges. Phys. Chem. 97, 1371 (1993). [42] W.J. Jones, US Patent 3,440,882, Thermometer, 1966. [43] H. Seto, M. Ueda, and H. Segawa, US Patent 3,704,625, Thermometer Using Liquid Crystal Compositions, 1970. [44] D.J. Broer, J. Lub, and G.N. Mol, Nature 378, 467 (1995). [45] P.P. Crooker and D.K. Yang, Appl. Phys. Lett. 57, 2529 (1990). [46] R.A.M. Hikmet, Adv. Mater. 4, 679 (1992). [47] D.K. Yang, J.W. Doane, Z. Yaniv, and J. Glasser, Appl. Phys. Lett. 64, 1905 (1994). [48] A. Petri, Ch. BraÈuchle, H. Leigeber, A. Miller, H.-P. Witzel, and F.-H. Kreuzer, Liq. Cryst. 15, 113 (1993). [49] P. Pal¨y-Muhoray and K.D. Singer, Optics & Photonics News (Opt. Soc. Am.), Sept. 1995, 17 (1995). [50] W. Schlichting, S. Faris, L. Li, B. Fan, J. Kralik, J. Haag, and Z. Lu, Mol. Cryst. Liq. Cryst. 301, 771 (1997). [51] T.J. Buninng and F.-H. Kreuzer, Trends Polymer Sci. 3(10), 318 (1995). [52] A. Stohr and P. Strohriegl, Mol. Cryst. Liq. Cryst. 299, 211 (1997). [53] J.S. Patel and R.B. Meyer, Phys. Rev. Lett. 58, 1538 (1987). [54] J.J. Wysocki, J. Adams, and W. Haas, Phys. Rev. Lett. 20, 1024 (1968). [55] G.W. Taylor and D.L. White, US Patent 3,833,287, Guest±Host Liquid Crystal Device, 1973. [56] D.K. Yang and J.W. Doane, SID Dig. Tech. Papers 1992, 759 (1992). [57] G.P. Crawford and S. Zumer (Ed.), Liquid Crystals in Complex Geometries, Taylor & Francis, London, 1996.

1. Introduction

27

[58] A. Kilian and A. Sonnet, Z. Naturforsch. 50a, 991 (1995). [59] R.B. Meyer, L. LieÂbert, L. Strzelecki, and P. Keller, J. Phys. Lett. 36, L69 (1975). [60] J.W. Goodby et al., Ferroelectric Liquid Crystals, Gordon and Breach, Philadelphia, 1991. [61] L.M. Blinov and V.G. Chigrinov, Electrooptic E¨ects in Liquid Crystal Materials, Springer-Verlag, New York, 1994. [62] H. Bock and W. Helfrich, Liq. Cryst. 18, 387 (1995). [63] N.A. Clark and S.T. Lagerwall, Appl. Phys. Lett. 36, 899 (1980). [64] S. Garo¨ and R.B. Meyer, Phys. Rev. Lett. 38, 848 (1977). [65] B.I. Ostrovskii and V.G. Chigrinov, Sov. Phys. Crystallogr. 25, 322 (1980). [66] J.S. Patel, Appl. Phys. Lett. 60, 280 (1992). [67] A.N. Vtyurin, V.P. Ermakov, B.I. Ostrovskii, and V.F. Shabanov, Phys. Stat. Sol. B 107, 397 (1981). [68] J.-Y. Liu, M.G. Robinson, K.M. Johnson, D.M. Walba, M.B. Ros, N.A. Clark, R. Shao, and D. Doroski, J. Appl. Phys. 70, 3426 (1991). [69] K. Schmitt, R.-P. Herr, M. Schadt, J. FuÈnfschilling, R. Buchecker, X.H. Chen, and C. Benecke, Liq. Cryst. 14, 1735 (1993). [70] J. Ortega, C.L. Folcia, J. Etxebarria, M.B. Ros, and J.A. Miguel, Liq. Cryst. 23, 285 (1997). [71] M. Loddoch, G. Marowsky, H. Schmid, and G. Heppke, Appl. Phys. B 59, 591 (1994). [72] M. Schubert and B. Wilhelmi, Nonlinear Optics and Quantum Electronics, Wiley, New York, 1986. [73] I.-C. Khoo and S.-T. Wu, Optics and Non-linear Optics of Liquid Crystals, Series in Nonlinear Optics, Vol. 1, World Scienti®c, Singapore, 1993. [74] P. Pal¨y-Muhoray, in: Liquid CrystalsÐApplications and Uses (edited by B. Bahadur), Vol. 1, Chap. 18, World Scienti®c, Singapore, 1990. [75] T. Niori, T. Sekine, J. Watanabe, T. Furukawa, and H. Takezoe, J. Mater. Chem. 6, 1231 (1996). [76] G. Heppke and D. Moro, Science 279, 1872 (1998). [77] D.R. Link, G. Natale, R. Shao, J.E. Maclennan, N.A. Clark, E. KoÈrblova, and D.M. Walba, Science 278, 1924 (1997). [78] S. Diele, G. Pelzl, and W. Weiss¯og, Liquid Crystals Today 9, 8 (1999). [79] R. Macdonald, F. Kentischer, P. Warnick, and G. Heppke, Phys. Rev. Lett. 81, 4408 (1998). [80] D. Bennemann, G. Heppke, A.-M. Levelut, and D. LoÈtzsch, Mol. Cryst. Liq. Cryst. 260, 351±360 (1995). [81] U. Seifert, Adv. Phys. 46, 13 (1997). [82] J.-M. Lehn, Supramolecular Chemistry, VCH, Weinheim, 1995.

2

Classroom Experiments with Chiral Liquid Crystals Pawel Pieranski

2.1

About Classroom Experiments

``I have never understood why it is so often assumed by university lecturers that their students are so highly motivated that the usual rules for retaining the attention of an audience, for making the subject exciting, for introducing the elements of humor and drama, can all be suspended.'' This citation from ``The Art and Science of Lecture Demonstration'' by Charles Taylor, that I found in the American Journal of Physics, is a perfect motto for this chapter. Indeed, having had long experience of lectures on liquid crystals, I am convinced that classroom experiments can be widely used for the bene®t of students and the lecturer. I tried to share this conviction with other physics lecturers and a few years ago, during a meeting of the French Physical Society in Marseille, I presented a collection of simple classroom experiments on nematic and smectic liquid crystals. This presentation has been published in the Bulletin de l'Union des Physiciens [1]. It seems to have incurred some positive response because a number of high-school and university teachers contacted me and asked for additional information about the experiments, with a view to realizing teaching projects with their own students. Let me mention that the experiment which has been appreciated most was the one with the nematic prism, that has been shown to me during my stay in Professor Heppke's laboratory in Berlin and which I used with success in my lectures as an introduction to nematic liquid crystals. My aim here will be to extend this collection of classroom experiments to chiral liquid crystals.

2.2

Twisted Nematic Display

As the ®rst experiment of this new series, the twisted nematic display is the most appropriate because, on the one hand, the explanation of its operational principle is based on properties of the nematic phase which are usually 28

2. Classroom Experiments with Chiral Liquid Crystals

29

Figure 2.1. Schema of the 16 pixel nematic display.

discussed at the beginning of a course on liquid crystals, and on the other hand, the concept of the chirality can be introduced by the example of the twisted texture. Let us start with a description of the experimental set-up, which was built by two students who were experimenting in my laboratory. The scheme of the 16 pixel display is shown in Figures 2.1 and 2.2. The display is made up of two rectangular glass plates assembled in a thin sandwich by means of two mylar spacers 200 mm thick and a few droplets of an epoxy glue placed at the edges of the cell. The internal sides of each of the glass plates contain four stripe-shaped ITO electrodes that have been etched from an initially uniform ITO coating. The etching process, which I learned from R. Meyer, is very simple: (1 ) stripes of an electric adhesive tape are used to protect the areas of the coating corresponding to the electrodes; (2 ) the glass plate with protected electrodes is immersed in a shallow layer of a 10% solution of hydrochloric acid in water; (3 ) zinc powder is poured in the solution above the glass plate; (4 ) the hydrogen gas ``in statu nascendi'' reduces the nonprotected ITO coating which is dissolved; and

30

P. Pieranski

Figure 2.2. Perspective view of the 16 pixel twisted nematic display.

(5 ) the glass plate is rinsed with water and the protective adhesive tape is removed. After the etching process, the glass surfaces have to be covered with a thin layer of polyvinylic alcohol for the nematic anchoring. This is done simply by immersing the plates in a water solution of polyvinylic alcohol (1%) and then slowly pulling them out of the solution. After drying, the surface of the glass plate is ready for rubbing. This can be done with a pharmaceutical

2. Classroom Experiments with Chiral Liquid Crystals

31

Figure 2.3. The helical (twisted) con®guration of the director ®eld imposed by the orthogonal anchoring conditions on the limit glass plates.

cotton. The rubbing direction has to be parallel to the direction of the stripeshaped electrodes. To ®ll the display cell, any commercially available nematic eutectic mixture (e.g., E9 from Merck) for twisted liquid crystal displays can be used. The liquid crystal has to be introduced slowly into the gap between the glass by means of a pipette. After several minutes of annealing, necessary for the elimination of disclinations and relaxation of the director ®eld, the display is ready for operation. The equilibrium con®guration of the director ®eld, satisfying the boundary conditions on the limit glass plates, is the helical one n ˆ ‰cos…pz=d†; sin…pz=d†; 0Š:

…2:1†

As shown in Figure 2.1, two ac voltages in phase opposition have to be used. The horizontal (line) electrodes can be connected by a system of four switches either to the ground or to the ®rst ac voltage Vline ˆ Va cos…ot†. The vertical (column) electrodes can be connected by the second system of switches to the ground or to the second ac voltage Vcol ˆ ÿVa cos…ot†. As a consequence, the voltage di¨erence at each crossing of electrodes (that is to say, at each pixel) can be 0Ðstate 0 in Figure 2.1, Va cos…ot†Ðstate 1, or 2Va cos…ot†Ðstate 2. Let Vc be the threshold voltage for the Fredericks transition v u uK ‡ K3 ÿ 2K2 t 1 4 : …2:2† Vc ˆ p eo ea If Va Va p < Vc < 2 p ; 2 2

…2:3†

then the helical con®guration of the director is altered only in pixels in state 2. Let us consider now the optical action of pixels in states 0 or 1 where the helical con®guration is preserved. As pointed out in Annex 1, when the

32

P. Pieranski

wavevector k is parallel to z and l f d, then two normal modes with linear polarization can propagate in the system: (1 ) the extraordinary mode which locally is polarized in the direction parallel to n…z†: E e ˆ En…z†e i…ke
rÿot† ;

…2:4†

(2 ) the ordinary mode which is polarized in the direction m perpendicular to n…z†: E o ˆ Em…z†e i…ko
rÿot† ;

…2:5†

where ko ˆ

2pno z l

and

ke ˆ

2pne z l

…2:6†

are wave vectors de®ned by the ordinary no and extraordinary ne indices of refraction. Equations (2.4) and (2.5) indicate clearly that the polarization of light follows the director in its helical con®guration. In the case of crossed polarizers, this means that pixels in the states 0 or 1 are transmitting light. For the two pixels in state 2, the helical con®guration is distorted (see Figure 2.4(b)) and the light polarization does not follow the director ®eld any longer. In the limit of the voltage Va much larger than Vc , the director in the bulk of the cell is aligned along the electric ®eld; nkz. In this limit, and for propagation in the z-direction, the cell behaves as an isotropic medium and light polarization is not a¨ected. As a result, when the polarizers are crossed, the two pixels in state 2 appear as black.

Figure 2.4. The director con®guration for (a) weak and (b) strong electric ®elds.

2. Classroom Experiments with Chiral Liquid Crystals

33

Figure 2.5. The optical action of the display is reversed when the polarizers are (b) parallel instead of (a) being crossed.

During demonstrations with our 16 pixel display, it is interesting to show that its optical action is reversed when the polarizers are parallel instead of being crossed (Figure 2.5). Another interesting feature of the display is its relaxation time which is of the order of 15 s. Such a long relaxation time (much longer than in real TN displays) is due to the thickness of the cell. Indeed, the relaxation time in the Fredericks transition varies as d 2 , so that when the thickness of the cell is 20 mm instead of 200 mm, the relaxation time becomes 0.15 s instead of 15 s.

2.3

Measure of the Cholesteric Pitch Using the Cano Wedge

In the twisted nematic display, the helical con®guration of the director ®eld is due to the crossed anchoring conditions on the limit glass plates. Let us note that instead of the right-hand helix shown in Figure 2.3, the passage from the anchoring direction parallel to x on the bottom glass plate, to the anchoring direction parallel to y on the upper plate, could be realized through a left-hand helix. In the nematic phase, the right and left helix con®gurations must be identical energetically, because the whole system is invariant with respect to the mirror re¯ection symmetry. In the cholesteric phase, the helical con®guration of the director is the consequence of the molecular chirality. From the microscopic point of view, unlike in the nematic phase where the molecules have a tendency to be locally parallel to each other, in cholesteric materials, the adjacent molecules have a tendency to form a small angle as shown in Figure 2.6. As we will point out in Section 2.6, such a local tendency cannot be satis®ed globally and the system is frustrated.

34

P. Pieranski Figure 2.6. The adjacent chiral molecules have a tendency to form a small angle.

In the cholesteric phase, this local tendency of molecules to twist is satis®ed in only one direction of spaceÐlet us say the z-axis as shown in Figure 2.7. The pitch p of such a ``natural'' helix is usually quite short in pure materials: p A 1 mm and cannot be deduced from direct observations in a polarizing microscope. In order to measure the pitch, the so-called Cano wedge method [2] can be used. It is very tricky and quite simple to realize. These are two reasons at least for which the Cano wedge experiment deserves to be presented here. In this experiment, the liquid crystal is introduced into a gap between two glass surfaces treated for the planar anchoring. This anchoring can be achieved using the rubbed polymer ®lm (see the previous section) or SiO ®lm evaporated under the oblique incidence. In the geometry shown in Figure 2.8, one surface is plane (e.g., a glass plate) and the second one is cylindrical

Figure 2.7. The helical con®guration of molecules in the cholesteric phase.

2. Classroom Experiments with Chiral Liquid Crystals

35

Figure 2.8. The geometry of the Cano wedge experiment.

Figure 2.9. Photograph of the cholesteric texture in the Cano wedge.

Figure 2.10. The texture of a cholesteric liquid crystal in the Cano wedge: the number of half-pitches located in the gap varies as a function of the thickness h…x†.

(e.g., a cylindrical lens) but a plane/plane geometry can also be used. As a liquid crystal, a mixture of a nematic (e.g., E9 from Merck) with a cholesterogene (e.g., CB15 from Merck) can be used. The pitch p of this mixture depends on the concentration c of CB15 in E9. It is in®nite for c ˆ 0 and of the order of 1 mm for c ˆ 30%. On the photograph taken in the transmitted light in the polarizing microscope, one observes a system of parallel bands separated by lines (Figure 2.9). In this texture, the number of half-pitches located in the space between two plane surfaces varies from one band to another. The corresponding texture is shown in Figure 2.10.

36

P. Pieranski

Figure 2.11. This diagram shows variation of the distortion energy with the coordinate x in the plane/plane geometry.

The explanation of this texture is very simple. Let h…x† be the local thickness. Only for discrete values xn of the coordinate x, the local thickness h…xn † corresponds exactly to an integer number n of half-pitches located in the gap p …2:7† hn ˆ n : 2 Everywhere else, the thickness is either too large or too small. As a consequence, the cholesteric texture is distorted and an elastic energy is involved. For a given number n of half-pitches, the distortion energy density becomes   2p np 2 1 ÿ : …2:8† fn …x† ˆ 2K2 p h…x† For the case of plane/plane geometry, this energy is plotted versus x for several values of n on the diagram of Figure 2.11. It is obvious that in order to minimize this distortion energy, the number n of the half-pitches must vary from band to band. In order to ®nd the position xnÿ1; n of the boundary between two adjacent bands, one has to solve the equation …2:9† fnÿ1 …xnÿ1; n † ˆ fn …xnÿ1; n † or, explicitly,

The solution is

  2p …n ÿ 1†p 2p np ÿ ÿ ˆÿ : p h…xnÿ1; n † p h…xnÿ1; n †

…2:10†

p h…xnÿ1; n † ˆ …2n ÿ 1† : 4

…2:11†

2. Classroom Experiments with Chiral Liquid Crystals

37

Figure 2.12. Disclination w between two bands in the Cano wedge texture.

In the plane/plane geometry, one has h ˆ ax where a is the wedge angle and one gets p …2:12† xnÿ1; n ˆ …n ÿ 12† : 2a Knowing the wedge angle a, the pitch p can be determined from the plot xnÿ1; n versus n. Let us note at the end of this section, that at the boundary between two adjacent bands, in the Cano wedge texture, one ®nds a w disclination of the cholesteric texture. This disclination is a topological necessity because the director makes a p turn on the Burgers circuit shown in Figure 2.12.

2.4

Unwinding of the Cholesteric Pitch

The Cano wedge method is very convenient for measurements of variations of the cholesteric pitch in the presence of the magnetic ®eld orthogonal to the helix (Figure 2.13). The idea of this experiment, due to de Gennes [3] and

Figure 2.13. Unwinding of the cholesteric helix by the magnetic ®eld.

38

P. Pieranski Figure 2.14. Con®guration of the cholesteric helix in a large magnetic ®eld.

Meyer [4], is the following one. In the presence of the magnetic ®eld, the cholesteric helix is distorted: instead of a uniform rotation around the z-axis, there are regions in the texture where the director ®eld is almost parallel to the magnetic ®eld and these regions are separated by walls of thickness x in which the director makes a p turn (Figure 2.14). When the ®eld increases, these walls are more and more localized and the distances p=2 between them become larger and larger. This means that the cholesteric pitch p increases with the ®eld. There exists a critical ®eld Bc for which the pitch p diverges. For B > Bc , the walls disappear and the system becomes nematic. In order to ®nd the variation of the pitch with the ®eld and critical ®eld, one has to look for the minimum of the total energy per unit area of the system # 2 … "  1 qj wa 2 2 B sin j ; …2:13† ÿ qo ÿ dz K2 FA ˆ mo 2 qz where wa ˆ wk ÿ w? is the di¨erence between magnetic susceptibilities measured with the ®eld B parallel and perpendicular to the molecules. The Euler equation of this variational problem is x2

d 2j ‡ sin j cos j ˆ 0; dz 2

where x is the magnetic coherence length de®ned as s 1 mo K2 : xˆ wa B The ®rst integral of this equation is  2 1 2 dj ‡ sin 2 j ˆ 2 ; x dz k

…2:14†

…2:15†

…2:16†

2. Classroom Experiments with Chiral Liquid Crystals

39

where k is a constant related to the pitch p of the helix by the expression …p p dz ˆ dj ˆ 2xkK…k† …2:17† 2 0 dj and K…k† is the ®rst-order elliptic integral … p=2 dj q : K…k† ˆ 0 1 ÿ k 2 sin 2 j

…2:18†

In order to calculate k as a function of B, one has to minimize the energy FA with respect to k. As a result one gets pk B~ ˆ …qo x†ÿ1 ˆ ; 2E…k† where B~ is the dimensionless ®eld and … p=2 q 1 ÿ k 2 sin 2 j dj E…k† ˆ 0

is the second-order elliptic integral. From (2.17) and (2.19) one gets  2 p 2 ˆ K…k†E…k†: po p

…2:19†

…2:20†

…2:21†

Equations (2.19) and (2.21) give the variation of the pitch p versus the ®eld B in parametric form. From (2.21) one ®nds that the pitch p diverges when the integral K…k† diverges which takes place for k ˆ 1. Then from (2.19) one ®nds the critical ®eld s p mo K2 ; …2:22† Bc ˆ q0 wa 2 which varies as pÿ1 o , as expected. Results of experiments performed by Durand et al. [5] are presented in Figure 2.15. They show excellent agreement with the theory. These authors have also veri®ed that the critical ®eld varies as the inverse of the pitch po . The helical unwinding can also be observed when an electric ®eld is applied to a cholesteric liquid crystal with a positive dielectric anisotropy [6].

2.5

Selective Bragg Re¯ection from a Cholesteric Liquid Crystal

Many years ago, when the activity on liquid crystals was expanding in France under the guidance of de Gennes, my thesis director, Etienne Guyon, started to introduce some elements of liquid crystal physics into his lectures.

40

P. Pieranski

Figure 2.15. Variation of the cholesteric pitch versus the magnetic ®eld measured by the method of the Cano wedge by Durand et al. [5]. The liquid crystal used in this experiment was the mixture of PAA with cholesteryl chloride at concentration 0.02%. The continuous line corresponds to the theoretical prediction by de Gennes.

Selective Bragg re¯ection of light from cholesteric liquid crystals was one of his preferred lecture demonstrations. For this purpose, he was using a mixture of cholesteryl esters prepared by Lionel Liebert, one of the chemists of our laboratory. The composition of this mixture (75% cholesteryl nonanoate with 25% cholesteryl chloride) was chosen in a way that, on one hand, the range of the cholesteric phase was extended to the ambient temperature and, on the other hand, the Bragg re¯ection was in the visible wavelength range. Such a sample, contained in a test tube and o¨ered for experimentation to students, presented beautiful bright pearly colors changing slightly when the liquid crystal was ¯owing in the test tube. In order to record the spectrum of light re¯ected from the cholesteric liquid crystal, it is better to use a thin slab of the same mixture con®ned between a microscope glass slide and a cover glass. In such a sample, the orientation of the liquid crystal can be e½ciently improved by a gentle backand-forth gliding motion of the cover glass. After this action, the cholesteric helix is orthogonal to the slab. Bragg re¯ections of light from these cholesteric samples are obviously due to the periodic structure of the cholesteric phase, and the wavelength of the re¯ected light is determined by the pitch p of the helix. Using circular

2. Classroom Experiments with Chiral Liquid Crystals

41

Figure 2.16. Samples for demonstration of the Bragg re¯ection from cholesteric liquid crystal.

polarizers, it is easy to show that the re¯ected and transmitted light are circularly polarized: the re¯ected and transmitted beams have opposite handedness. This circular polarization of the Bragg re¯ection is due to the chiral structure of the cholesteric helix. The detailed analysis of light propagation in the cholesteric helix is quite complex. It consists in the search for eigenmodes of Maxwell equations in a medium with the position-dependent dielectric permittivity tensor. In order to obtain the explicit expression of eij , the uniaxial approximation of the cholesteric phase can be adopted for the purpose of simplicity. In this approximation, the cholesteric phase is supposed to be locally uniaxial like the nematic phase, and e? and ek are dielectric constants measured in directions perpendicular and parallel to the local director. In terms of n, the dielectric tensor of the nematic phase can be written as eij ˆ e? dij ‡ …ek ÿ e? †ni …z†nj …z†:

…2:23†

Let us now suppose that the director n…z† forms a right-hand helix (Figure 2.7): nx ˆ cos…qz†;

ny ˆ sin…qz†;

nz ˆ 0;

q ˆ 2p=p:

…2:24†

Using (2.23), one gets 0e ‡ e ? k B 2 B e^…z† ˆ B @ 0 0

0 ek ‡ e? 2 0

1

0 cos…2qz† sin…2qz† C e ÿe ? k C @ sin…2qz† ÿcos…2qz† ‡ 0 C A 2 0 0 e? 0

1 0 0 A: 0 …2:25†

Let us note that this expression has two parts. The ®rst one is positionindependent and represents the dielectric tensor eij averaged on a volume with dimensions much larger than the pitch p. It is worthwhile to emphasize that this averaged tensor is uniaxial with the symmetry axis parallel to the helix. Moreover, its global anisotropy is opposite to the local anisotropy. For example, if the local dielectric anisotropy is negative, that is to say, if ek < e? , then

42

P. Pieranski

e? >

e? ‡ ek : 2

…2:26†

Due to this property, the cholesteric phase can be aligned by the electric ®eld with the helix parallel to the direction of the ®eld. The second part of the expression (2.25) represents a purely biaxial (traceless) tensor rotating along the z-axis. The period of rotation is p=2 because the wavevector is 2q. This means that wavelength l of the re¯ected light is related to p=2 and not to p by the Bragg relation l 2… p=2† cos Q ˆ : n

…2:27†

The refractive index n in this formula is an average between the ordinary and extraordinary indices de®ned by ne2 ˆ ek and no2 ˆ e? , and the wavelength l corresponds to the center of the re¯ection peak. In reality, as shown in Annex II where Maxwell equations are solved exactly, the Bragg re¯ection occurs in quite a large band of wavelengths (analog to the Darwin band in X-ray crystallography) de®ned by pno < l < pne :

…2:28†

Because of the large optical anisotropy of liquid crystals, the relative width Dl=l of this Darwin band is much larger than in X-ray crystallography where the modulation of the refractive index (for X-rays) is of the order of 10ÿ6 . Experimentally, the measure of the Darwin band in cholesteric liquid crystals requires production of perfectly aligned samples and the use of a spectrometer. This can hardly be done during a lecture. The experimental classroom demonstration of the circular polarization of the re¯ected light beam is much easier so it is pertinent to explain its theoretical origin. The simplest explanation of this phenomenon has been proposed by de Gennes [7]. As a starting point he took the formula a ˆ f
e^…q†
i

…2:29†

used for explanation of the light scattering from ¯uctuations of the director in nematic liquid crystals. In this expression, a is a factor proportional to the scattered light intensity, eij …q† is the amplitude of the Fourier component with wavevector q of the dielectric tensor ¯uctuation, while vectors f and i are polarizations of, respectively, re¯ected and incident light waves. In the context of Bragg re¯ection from the cholesteric helix, we know already from the expression (2.25) that there is just one Fourier component with wavevector 2q. Its amplitude is complex because the second term in the expression (2.25) can be written as 0 1 2 3 1 ÿi 0 ek ÿ e? @ ÿi ÿ1 0 Ae i 2qz 5: …2:30† Re4 2 0 0 0

2. Classroom Experiments with Chiral Liquid Crystals

43

Figure 2.17. Selection rules for the re¯ection and transmission of light by the cholesteric liquid crystal.

We are interested here in light scattering at normal incidence (light propagation along the helix direction) so that vectors f and i have only x and y components. Let the incident wave have the right circular polarization represented schematically in Figure 2.17: E in ˆ E Re‰…1; i; 0†e i…otÿkz† Š:

…2:31†

In such a case, i ˆ …1; i; 0†. Let us now calculate the product e^…2q†
i. One gets e^…2q†
i ˆ 2…1; i; 0†:

…2:32†

44

P. Pieranski

In order to obtain real a, the polarization vector f must be of the form f ˆ ‰1; ÿi; 0Š so that the re¯ected wave is E ref ˆ E Re‰…1; ÿi; 0†e i…ot‡kz† Š:

…2:33†

As shown in Figure 2.17, this re¯ected wave has the same polarization as the incident waveÐthe right circular one. Now, if the incident wave had the left circular polarization, the product e^…2q†
i would be zero which means that this wave is not scattered but transmitted by the cholesteric slab.

2.6

Kossel Diagrams of Blue Phases [8]

``They are totally useless, I think, except for one important intellectual use, that of providing tangible examples of topological oddities, and so helping to bring topology into the public domain of science, from being the private preserve of a few abstract mathematicians and particle theorists.'' This sentence, due to Sir Charles Frank, about blue phases is paradoxically one of the arguments in favor of presentation of experiments with blue phases here. Indeed, the structures of blue phases can be considered as very tricky solutions, di¨erent from one of the cholesteric phase, of the problem of the geometrical frustration in chiral systems mentioned in Section 2.3. The second argument for classroom experiments with blue phases is the historical one; blue phases were probably those of the ®rst liquid crystal phases observed by the German physicist Otto Lehmann at the beginning of twentieth century. As to the choice of Kossel diagrams, from many possible and interesting experiments with blue phases, it is justi®ed by the opportunity that it gives the lecturer to introduce the most fundamental concepts of crystallography: Bravais lattice, reciprocal lattice, representation of the crystal structure as a Fourier series, selection rules due to the symmetry of the crystal, etc. Finally, the explanation of Kossel diagrams of blue phases, based on the theory elaborated by Dick Hornreich et al. [10], reminds us of this wonderful man, our friend. At the beginning of this chapter, it seemed necessary to emphasize that experiments with blues phases were certainly more di½cult than those with the nematic and cholesteric phases, mainly because a very precise temperature regulation must be used. Indeed, blue phases exist in a very narrow temperature interval between the isotropic and cholesteric phases. This di½culty is partly removed if mixtures such as E9 with CB15 are used instead of pure materials, because in mixtures the blue phases can occur at temperatures around 30  C while in pure materials the typical temperature range is about 100  C. Typically, the mixture of 50% of CB15 in E9 is a good choice, because it possesses a quite low-temperature phase sequence Chol

34:3  C

BPI

34:6  C

BPII

35:7  C

Isotropic

2. Classroom Experiments with Chiral Liquid Crystals

45

Figure 2.18. Cell for studies of blue phases: (a) general view, (b) view of a Blue Phase I crystal grown in the isotropic phase and oriented with (110) planes parallel to the glass surface.

The experimental set-up for studies of Kossel diagrams is depicted in Figures 2.18 and 2.19. The liquid crystal sample is contained in the space of thickness about 0.2 mm between a thin cover glass and a glass cylinder of diameter of 3 mm (Figure 2.18). The temperature of this cell is regulated by means of Peltier elements. This sample is in thermal contact through the

Figure 2.19. Schema of the optical set-up for studies of Kossel diagrams.

46

P. Pieranski Figure 2.20. Principle of the Kossel diagrams.

immersion oil with a high aperture objective …40†. For this reason, the temperature of the objective is regulated too by an independent system of Peltier elements. A small vertical temperature gradient that can be produced, thanks to this system of two independent temperature regulations, is very useful for a controlled growth of Blue Phase II crystals from the isotropic phase. The objective ®xed on a re¯ecting microscope is used for illumination of the sample with the beam from a monochromator and for production of Kossel diagrams. Theoretically, the ideal con®guration for production of the Kossel diagrams would consist of a monochromatic point light source situated inside the crystal and of a spherical screen surrounding the crystal. In such a con®guration, any family (hkl ) of crystal planes satisfying the Bragg relation 2dhkl cos y ˆ l would produce on the screen a Kossel ring centered around the normal to the crystal planes. The symmetry of the whole set of such rings re¯ects the symmetry of the crystal. In our experiment, the immersion objective produces a cone of light illuminating the crystal and receives rays re¯ected by crystal planes (Figure 2.20). The re¯ected rays are focused by the objective into a Kossel ring in the focal plane. Due to the ®nite aperture of the objective, only a part of the Kossel diagram is visible so that in order to compose the whole Kossel diagram it is necessary to combine images obtained with di¨erent orientations of the crystal. The typical images of Kossel diagrams of the Blue Phase II monocrystals with [100], [110], and [111] axes parallel to the optical axis of the system are shown in Figures 2.21(a), (b), and (c) [8]. The fourfold, threefold, and two-

2. Classroom Experiments with Chiral Liquid Crystals

47

Figure 2.21(a). Kossel diagram of the Blue Phase II seen along the fourfold axis [100]. The four Kossel lines visible on the experimental image (left) obtained with l ˆ 373 nm, are identi®ed with (110), (1-10), (101), and (10-1) lines on the theoretical diagram. Let us note that the (100) line from the theoretical diagram is invisible on the di¨raction image because the aperture of the objective is not large enough.

Figure 2.21(b). Kossel diagram of the Blue Phase II seen along the threefold symmetry axis [111]. The apparent sixfold symmetry of the experimental diagram is due to the re¯ection of light on the surface of the cylinder glass in the cell. Due to this re¯ection the diagrams from the ``north'' and ``south'' hemispheres are superposed.

Figure 2.21(c). Kossel diagram of the Blue Phase II seen along the twofold axis [110]. Let us note that the (100), (010), and (110) lines have common intersection points.

48

P. Pieranski

Figure 2.22. The unit of the Blue Phase II is cubic and it contains about 10 7 molecules.

fold symmetries of these Kossel diagrams are obvious so that the cubic structure of the Blue Phase II is clearly demonstrated. Besides the determination of the Bravais lattice type, more information can be extracted from these diagrams. In order to do this, it is necessary to understand how Kossel lines are formed and what their relationship is with the detailed structure of the Blue Phase II. Let us start with a few simple explanations about the blue phase structure. A much more detailed description is given by Peter CrookeÂr in Chapter 7. We know already that the Blue Phase II is made of cubic unit cells and that the lattice constant must be of the order of 0.2±0.4 mm because the diffraction diagrams are obtained with the visible light. Such a huge unit cell contains about 10 7 molecules (Figure 2.22). These molecules are free to di¨use inside the cell, like in a liquid, but their orientation depends on the position in the cell. Physical properties of blue phases are due to such a structure. From the point of view of the light di¨raction, the pertinent quantity is the dielectric tensor eij …r† which is a periodic function of r which, therefore, can be decomposed into a Fourier series. This decomposition was done for the ®rst time by Dick Hornreich et al. [10] who used the traceless part of the dielectric tensor as the order parameter for a Landau-type theory of blue phases. The traceless part of the symmetric dielectric tensor has ®ve independent coe½cients. For this reason, for each q…hkl† vector of the reciprocal lattice, ®ve independent Fourier components have to be considered. We already know one of them from the section on cholesteric liquid crystals (2.25). Let

2. Classroom Experiments with Chiral Liquid Crystals

49

2p/q

(a)

(b)

(c)

(d)

(e)

Figure 2.23. Graphical representation of the ®ve independent Fourier components of the dielectric tensor for a given wavevector q…hkl† from the reciprocal lattice.

us write it here once more like this 0

cos…qz† e^2 …hkl† ˆ e2 …hkl†@ sin…qz† 0

1 sin…qz† 0 ÿcos…qz† 0 A: 0 0

…2:34†

This Fourier component corresponds to a purely biaxial tensor forming a right-hand helix along the z-axis that has been chosen along the direction of the reciprocal lattice vector q…hkl† (see Figure 2.23(b)). The second Fourier component is similar except for the sense of the helix which is opposite 0 1 cos…qz† ÿsin…qz† 0 …2:35† e^ÿ2 …hkl† ˆ eÿ2 …hkl†@ ÿsin…qz† ÿcos…qz† 0 A: 0 0 0 It is represented graphically in Figure 2.23(a). The third and fourth Fourier components also correspond to helices formed by a rotating purely biaxial tensor but the tensor now has a di¨erent orientation with respect to the helix axis (Figures 2.23(c) and (d)): 0 1 0 0 cos…qz† 0 sin…qz† A …2:36† e^1 …hkl† ˆ e1 …hkl†@ 0 cos…qz† sin…qz† 0 and

0

0 e^ÿ1 …hkl† ˆ eÿ1 …hkl†@ 0 cos…qz†

1 0 cos…qz† 0 ÿsin…qz† A: ÿsin…qz† 0

…2:37†

50

P. Pieranski

Finally, the ®fth Fourier component corresponds to a uniaxial tensor with its symmetry axis parallel to q…hkl† and whose amplitude is modulated as a function of z (Figure 2.23(e)): 0 1 ÿsin…qz† 0 0 A: 0 ÿsin…qz† 0 …2:38† e^0 …hkl† ˆ e0 …hkl†@ 0 0 2 sin…qz† In conclusion, the traceless part of the dielectric tensor can be written as the following series: e^…r† ˆ

2 X X

e^m …qhkl †;

…2:39†

hkl mˆÿ2

where qhkl ˆ

2p …h; k; l† a

…2:40†

are vectors of the reciprocal lattice. The method of Kossel diagrams allows us to detect what Fourier components are present in this series and by this means to determine the detailed symmetry (space group) of the crystal. Today, we know that the Blue Phase II phase has the O 2 symmetry. In this space group, there is a 42 screw axis (rotation of 90 followed by translation a=2) in the direction [100]. This symmetry element introduces a selection rule for em …h00† components. For example, when …hkl† ˆ …100†, a glance at Figure 2.23 tells us that the m ˆ 1, ÿ1, and 0 components must vanish because they are not invariant under the action of the 42 screw axis. On the contrary, the m ˆ 2 and ÿ2 helices in Figures 2.23(a) and (b) transform into themselves when they are rotated by 90 and shifted by a half-wavelength. For a (200) wave vector, only the m ˆ 0 component satis®es this symmetry condition. For a (110) wave vector, one has to check the invariance with respect to a two-fold axis. Once again, it is evident from Figure 2.23 that only m ˆ ÿ2; 0, and 2 components are invariant. Finally, because of the threefold symmetry axis in the [111]-direction, only the m ˆ 0 component can be present for the (111) wave-vector. After this introduction to the representation of blue phase structures, in terms of the Fourier decomposition of the dielectric tensor, let us come back to the discussion of Kossel diagrams in Figure 2.21. Let us suppose that some em …hkl† Fourier component is present in the crystal and that the q…hkl† vector is directed along the optical axis of the microscope. As shown in Figure 2.22, all rays incident on the …hkl† plane at the Bragg incidence y will be re¯ected. These rays form a cone of angle y around the direction q…hkl† and are focused in the focal plane of the objective into a ring. When the q…hkl† vector is oblique with respect to the optical axis, only a part of the Kossel ring is visible in the focal plane of the objective.

2. Classroom Experiments with Chiral Liquid Crystals

51

Using this working principle, all Kossel lines present in the diagrams of Figure 2.21 can be identi®ed. In Figure 2.21(a), four (110) lines are visible. One can check that these lines have a circular polarization. This means that among three possible components …m ˆ ÿ2; 0; 2†, one …m ˆ 2†, has an amplitude much larger than the other two. This property has been explained by Dick Hornreich et al. in terms of the Landau-type theory. It results from the fact that the cholesteric liquid crystal has a well-de®ned chiralityÐS (lefthand) or R (right-hand). If the chirality is R, then the m ˆ 2 component is energetically more favorable than the other two. In Figure 2.21(a), the (100) line which should form a central ring is absent because the Bragg angle is larger than the aperture of the objective. In order to observe the (100)-like lines, one has to choose crystals with di¨erent orientations. This is the case in Figures 2.21(b) and (c) where Kossel diagrams are oriented with the [111] and [110] axes parallel to the optical axis of the microscope. In Figure 2.21(c), the lateral lines are segments of the (100) and (010) Kossel rings of large diameter. These Kossel lines have the circular polarization which means once again that the m ˆ 2 component has a much larger amplitude than m ˆ 0 and m ˆ ÿ2. Let us emphasize at the end of this section that the method of Kossel diagrams has also been used with success for studies of the SmC* [9].

2.7

SmC* Free-Standing Films

The discovery of the ®rst liquid crystal phase possessing the spontaneous polarizationÐthe SmC* phaseÐis one of the most exciting events in the history of liquid crystals from both experimental and theoretical points of view. Today, I still remember the very simple experimental set-up made by Bob Meyer for the purpose of detecting the polarization of the SmC* phase. As shown in Figure 2.24, the liquid crystal is held between two glass plates separated by two wires acting as spacers and used for the creation of a dc electric ®eld. The liquid crystal is oriented with smectic layers parallel to glass plates so that the applied ®eld is orthogonal to the SmC* helix axis. Let z be the helix axis and p ˆ 2p=q its pitch. The polarization P forms a helix similar to the one formed by the director n in the cholesteric phase P…z† ˆ P‰cos…qz†; sin…qz†; 0Š:

…2:41†

The thickness of the sample is much larger than the helix pitch so that the spontaneous polarization averaged on z is zero. For the same reason, such a thick sample produces the conoscopic image which looks like that of the SmA phase: it consists of four extinction brushes forming a cross and of a system of concentric interference rings. Under the action of the electric ®eld, the helical con®guration is distorted and the conoscopic image is shifted in the direction orthogonal to the applied ®eld. When the direction of the ®eld is reversed, the conoscopic image is shifted in the opposite direction. Such an

52

P. Pieranski

> Figure 2.24. The experimental set-up used by R. Meyer for a demonstration of the spontaneous polarization of the SmC* in DOBAMBC [11].

e¨ect proves that the coupling between the ®eld and the con®guration of the liquid crystal is linear which is possible only when the spontaneous polarization is present. The detailed explanation of this experiment is more complicated because one must ®rst solve the problem of the helix deformation under the torques that the electric ®eld E exerts on the local polarization P…z† and then explain the observed change in the conoscopic image. In order to avoid these di½culties, the SmC* sample should be much thinner than the pitch p. This is precisely the case of free-standing smectic ®lms which can be composed of a few smectic layers so that the polarization direction can be taken to be independent of z. The experimental set-up for experiments with smectic ®lms is depicted in Figure 2.26. Smectic ®lms are spanned on a rectangular frame with two mobile edges (Figure 2.26(c)). The electric ®eld is created by a system of four wire electrodes forming a small square (Figure 2.26(b)). The electrodes are ®xed to a translation stage allowing us to position them very close to the ®lm. The frame with the ®lm is ®xed to the translation stage of a re¯ecting polarizing microscope. Thanks to the x, y motion of the frame, any portion of the ®lm can be positioned above the system of electrodes. The system of electrodes is supplied by two independent voltage sources so that the direction of a dc ®eld can be chosen arbitrarily or a rotating electric ®eld can

2. Classroom Experiments with Chiral Liquid Crystals

53

Figure 2.25. Starting from the concept of ¯exo-electricity, R. Meyer pointed out that if the molecules are arranged in a spring-like con®guration (b) the spontaneous polarization should be present in the system (c). Liebert, Strzelecki, and Keller have synthesized DOBAMBC, the chiral substance possessing the SmC phase. Because of the chirality, the tilt azimuthal angle j rotates around the normal to smectic planes (a). As a consequence, the molecules form the spring-like con®guration imagined by Meyer.

be created. The liquid crystal SCE4 from BDH can be used because this commercial mixture possesses the SmC* phase at room temperature. In order to give a convincing demonstration of the spontaneous polarization, one can show how a 2p disclination is deformed by the applied ®eld. Usually, when the frame is slowly opened, a 2p disclination is present in the ®lm for topological reasons. Indeed, the meniscus surrounding the ®lm exerts an anchoring action on the azimuthal direction of the tilt in smectic layers. Suppose that the tilt direction be orthogonal to the ®lm edge as shown in Figure 2.27. In such a case, a 2p disclination must be present inside the ®lm. Due to large dimensions of the ®lm …3 mm  3 mm†, the elastic interaction of the disclination with the ®lm edges is small so that the position of the disclination is arbitrary in the ®lm plane. Using the x, y motion of the frame, the disclination can be positioned above the system of electrodes. In the absence of the ®eld, the four extinction brushes characteristic of the 2p disclination, observed in the polarizing microscope, form a cross (Figure 2.28(a) shows the vicinity of the disclination center) which means that the director ®eld c ˆ …cos j; sin j† is in its equilibrium con®guration

54

P. Pieranski

Figure 2.26. Experimental set-up for studies of the response of smectic ®lms to the electric ®eld. In an SmC* ®lm, a 2p disclination is induced by anchoring conditions at the edges of the ®lm. Thermal ¯uctuations of such a disclination have been studied by Muzny and Clark [12].

2p Figure 2.27. In an SmC ®lm, a 2p disclination is induced by anchoring conditions at the edges of the ®lm.

2. Classroom Experiments with Chiral Liquid Crystals

55

Figure 2.28. 2p disclination in the SmC* ®lm (a) at zero ®eld, and its response (b), (c) to a dc and (d) ac electric ®eld.

(a)

(d)

(b)

(c)

j ˆ c;

…2:42†

where c is the angle of polar coordinates …r; c†. When the ®eld is applied, the cross-like con®guration of the extinction brushes is transformed into a 2p wall connected to the disclination as shown in Figure 2.28(b)). When the ®eld direction is reversed, this 2p wall is ®rst destroyed and then recreated on the opposite side of the disclination (Figure 2.28(c)). In equilibrium, the structure of this 2p wall results from the balance between the torque G ˆ P  E, exerted by the ®eld E on the polarization P, and the elastic torque. Far enough from the disclination and from the edge of the ®lm, the balance of torques can be written as K

d 2j ÿ PE sin j ˆ 0 dx 2

or as

x2

d 2j ÿ sin j ˆ 0; dx 2

…2:43†

where x 2 ˆ K=…PE† and x is the coordinate orthogonal to the wall. In the second form, the equation (2.43) is known as the sine±Gordon equation and has the soliton-like solution j…x=x† ˆ 4 arctan‰e x=x Š;

…2:44†

for which the azimuthal angle j passes from 0 to 2p when x=x varies from ÿy to ‡y. More precisely, this variation of j with x is localized in a band of width 4x around x ˆ 0 (see Figure 2.29). Now, when instead of the dc ®eld, an ac ®eld (A10 kHz) is applied, the torque G ˆ P  E due to the coupling between the polarization P and the ®eld E averages to zero. However, due to the anisotropy ea ˆ ek ÿ e? of the dielectric tensor, there exists an electric torque proportional to E 2 : G ˆ eo ea …c
E†…c  E† ˆ eo ea E 2 sin j cos j:

…2:45†

56

P. Pieranski Figure 2.29. Structure of the 2p wall created in an SmC ®lm by the action of the dc electric ®eld E.

The equation of the balance of torques now takes the form x2

d 2 …2j† ÿ sin…2j† ˆ 0; dx 2

…2:46†

with x 2 ˆ K=…eo ea E 2 †. It is obvious that the soliton-like solution (2.45) concerns 2j so that j varies only from 0 to p across the wall. For this reason two p walls are created on each side of the 2p disclination as shown in Figure 2.28(d). It is interesting to note that this equation is identical with (2.26) that was used in Section 2.4 for calculation of the equilibrium structure of the cholesteric helix in the magnetic ®eld. At the end of this section, let us mention that very interesting phenomena are occurring in SmC* ®lms (as well as in SmC ®lms) when they are submitted to a rotating electric ®eld in which the angular frequency o is so high that the director c cannot follow the ®eld. In this so-called asynchronous regime, the electric ®eld exerts an e¨ective (time-averaged) torque G proportional to E 2 . This kind of experiment, performed by Cladis et al. [13], shows that the disclination has a very strange behavior illustrated by the six photographs in Figure 2.30. The disclination visible in these photographs carries a solid inclusion (a dust particle) in its core. This inclusion exerts an anchoring action on the director c, similar to the one at the edges of the ®lm. As explained in [14], it is possible to introduce and block a di¨erence

2. Classroom Experiments with Chiral Liquid Crystals

57

Figure 2.30. Behavior of a 2p disclination submitted to a rotating electric ®eld [13].

Dj ˆ W p …W ˆ 0; G1; G2; . . .† between the azimuthal angle at the inclusion j…ri ; c† ˆ W p ‡ c and the ®lm edge j…R; c† ˆ c. In the absence of the external torque, the distortion ®eld has a logarithmic con®guration j…r† ˆ jns …r† ‡ Sc ˆ W p ln

…r=R† ‡ Sc …ri =R†

…2:47†

visible on photograph 1 in Figure 2.30 and satisfying the condition KDj ˆ 0 of zero elastic torque. When the external torque is applied, the con®guration of the disclination changes as shown in the sequence of photographs in Figure 2.30. In particular, Cladis et al. have found that for a large enough torque G, the disclination leaves the center and orbits around a target-like pattern. In the steady state, the director c in the center of the target rotates with a constant angular velocity W so that the angle j increases with a constant rate: j ˆ Wt. At the edge of ®lm, the angle j stays constant so that the di¨erence Dj between the center and the edge of the ®lm increases. In reality, due to its orbital motion, the disclination acts as a sink, absorbing the creation of the angle j in the target pattern. This remarkable phenomenon

58

P. Pieranski

Figure 2.31. Topological ¯ows: motion of dust particles reveal in SmC ®lms permanent ¯ows driven by the rotating electric ®eld in the presence of the 2p disclination.

certainly deserves to be presented during a lecture on the properties of the SmC phase. Let us note that this phenomenon of orbiting disclination takes place for torques G larger than a certain threshold value Gcrit . For G < Gcrit , the disclination stays in the center and the distortion ®eld around the disclination tends to a static (time-independent) con®guration. Nevertheless, the system is not in equilibrium but only in a steady state. Indeed, observation of small dust particles ¯oating on the ®lm shows that the static con®guration of the director ®eld c is accompanied by permanent ¯ows in the ®lm. As shown in Figure 2.31, the dust particles are circulating around the disclination with velocities depending on a distance r from the center. Chevallard et al. [14] have explained this surprising behavior as due to the action of the so-called Ericksen elastic stresses which in the presence of the disclination can generate body forces driving ¯ows.

References [1] [2] [3] [4] [5]

P. Pieranski, Bull. Union Physiciens 91, 161 (1997). R. Cano, Bull. Soc. France MineÂr. Cryst. 34, 333 (1967). P.-G. de Gennes, Solid State. Comm. 6, 163 (1968). R.B. Meyer, Appl. Phys. Lett. 12, 281 (1968). G. Durand, L. Leger, F. Rondelez, and M. Veyssie, Phys. Rev. Lett. 22, 227 (1969). [6] J. Wysocki, J. Adams, and W. Haas, Mol. Cryst. Liq. Cryst. 8, 471 (1969).

2. Classroom Experiments with Chiral Liquid Crystals

59

[7] P.-G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993. [8] B. JeÂroÃme, P. Pieranski, V. Godec, G. Haran, and C. Germain, J. Phys. 49, 837 (1988). [9] H.-S. Kitzerow, G. Heppke, H. Schmidt, B. JeÂroÃme, and P. Pieranski, J. Phys. 51, 2087 (1990). [10] H. Grebel, R.M. Hornreich, and S. Shtrikman, Phys. Rev. A 28, 1114 (1983). [11] R.B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, J. Phys. 36, L-69 (1975). [12] C. Muzny and N. Clark, Phys. Rev. Lett. 68, 804 (1992). [13] P.E. Cladis, P.L. Finn, and H.R. Brand, Phys. Rev. Lett. 75, 1518 (1995). [14] C. Chevallard, J.-M. Gilli, T. Frisch, I.V. Chikina, and P. Pieranski, Mol. Cryst. Liq. Cryst. 328, 589±594 and 595±611 (1999). [15] More experiments with liquid crystals are described in the book Les cristaux liquides by P. Oswald and P. Pierauski, Gordon & Breach, 2000. The present chapter is inspired by this book.

Annex I.

Light Propagation in a Twisted Nematic Cell

In order to ®nd modes of light propagation in the twisted nematic cell, we start by representing the twisted structure as a stack of uniaxial birefringent layers of thickness dz. In this stack, the optical axis rotates by dj ˆ q dz

…AI:1†

from one layer to another. In each uniaxial layer, the modes of light propagation have linear polarizations and their respective wave vectors are ke ˆ ne …2p=l†z

and

ko ˆ no …2p=l†z;

…AI:2†

where ne and no are the extraordinary and ordinary indices of refraction and z is the unit vector parallel to the z-axis. Let eo …z† and ee …z† be the unit vectors parallel to the electric ®eld of these modes. We will look for the modes of the whole stack in the form Let

E…z† ˆ a…z†eo …z† ‡ b…z†ee …z†:

…AI:3†

Ein1 ˆ E…z† ˆ a…z†eo …z† ‡ b…z†ee …z†

…AI:4†

be the electric ®eld of the light wave penetrating from below into the slab …z; z ‡ dz†. After the passage in the slab, the electric ®eld becomes Es1 ˆ a…z†eo …z† exp…ÿiko dz† ‡ b…z†ee …z† exp…ÿike dz†:

…AI:5†

In agreement with (AI.2), the ®eld penetrating into the next slab can be written as Ein2 ˆ E…z ‡ dz† ˆ a…z ‡ dz†eo …z ‡ dz† ‡ b…z ‡ dz†ee …z ‡ dz†:

…AI:6†

This ®eld has to be identical with the one given in (AI.5): Es1 ˆ Ein2 :

…AI:7†

60

P. Pieranski

Figure AI.1. A twisted nematic represented as a stack of uniaxial layers. The optical axis rotates by dj ˆ q dz from one layer to another.

Let us represent vectors eo …z† and ee …z† on the basis eo …z ‡ dz† and ee …z ‡ dz† of the next slab eo …z† ˆ eo …z ‡ dz† ÿ djee …z ‡ dz†

…AI:8†

ee …z† ˆ ee …z ‡ dz† ‡ djeo …z ‡ dz†:

…AI:9†

and

Using ®rst-order approximations of exponentials and of functions a…z ‡ dz† and b…z ‡ dz†, one gets da ˆ ÿiko a ‡ qb dz

…AI:10†

db ˆ ÿqa ÿ ike b: dz

…AI:11†

and

We look for solutions of this system of equations in the form a…z† ˆ ao exp…az†;

b…z† ˆ bo exp…az†;

…AI:12†

and as a result we have to solve a second-order equation a 2 ‡ i…ko ‡ ke †a ÿ ko ke ‡ q 2 ˆ 0

…AI:13†

whose determinant is Dˆÿ

4p 2 …Dn† 2 l2

ÿ

16p 2 ; p2

…AI:14†

with Dn ˆ ne ÿ no . In the limit l f pDn;

…AI:15†

2. Classroom Experiments with Chiral Liquid Crystals

61

the roots of (AI.13) are a‡; ÿ ˆ ÿiko; e :

…AI:16†

In conclusion, there are two modes of the propagation of light in the twisted nematic cell 1 : a…z† ˆ ao exp…ÿiko z†;

b…z† ˆ 0;

…AI:17†

b…z† ˆ bo exp…ÿike z†:

…AI:18†

and 2 : a…z† ˆ 0;

The electric ®eld of these two modes follows the twist of the director in the approximation l f pDn. This property is essential for the working principle of twisted nematic displays.

Annex II. Light Propagation in the Cholesteric Phase The exact theory of light propagation in the cholesteric phase is based on the Maxwell equations rot E ˆ ÿ rot B ˆ div B ˆ 0

qB ; qt

1 qD ; eo c 2 qt

and

div D ˆ 0:

…AII:1† …AII:2† …AII:3†

The di½culty in ®nding solutions of the Maxwell equations in the case of the cholesteric phase comes from the fact that the dielectric tensor eij linking ®elds D and E: Di ˆ eo eij Ej

…AII:4†

depends on r. In the reference frame …x; y; z† with the z-axis parallel to the axis of the helix, the explicit form of this tensor has been given in Section 2.5: 1 0 ek ‡ e? 0 1 0 0 cos…2qz† sin…2qz† 0 C e ÿe B 2 ? k C B ek ‡ e? @ sin…2qz† ÿcos…2qz† 0 A: ‡ e^…z† ˆ B 0 0 C A @ 2 2 0 0 0 0 0 e? …AII:5† For the purpose of simplicity, we will limit here the analysis to the case of light propagation in the direction of the cholesteric helix. In such a case, the electric ®eld is orthogonal to z and only its x and y components are di¨erent from zero

62

P. Pieranski

Ex ˆ Ex …z†eÿiot ;

…AII:6†

Ey ˆ Ey …z†eÿiot :

…AII:7†

In this situation, the ®rst two Maxwell equations give ez 

qE ˆ ioB qz

and

ez 

where ez is a unitary vector along z. Using (AII.8) and (AII.4) one gets  2 d 2 Ei o ˆ ÿ eij Ej ; 2 dz c

qB ÿio ˆ D; qz eo c 2

…AII:8†

i; j ˆ x; y:

…AII:9†

Knowing the expression (AII.5) of the dielectric tensor one obtains the following system of coupled equations for the amplitudes Ex …z† and Ey …z† of the electric ®eld

with ko2

ÿ

d 2 Ex ˆ ko2 Ex ‡ ka2 ‰cos…2qz†Ex ‡ sin…2qz†Ey Š; dz 2

…AII:10a†

ÿ

d 2 Ey ˆ ko2 Ey ‡ ka2 ‰sin…2qz†Ex ÿ cos…2qz†Ey Š; dz 2

…AII:10b†

 2 o ek ‡ e? ˆ c 2

ka2

and

 2 o ek ÿ e? ea ˆ k2 : ˆ c 2 2

…AII:11†

Using new variables EG ˆ Ex G iEy ;

…AII:12†

equations (AII.10) become ÿ

d 2 E‡ ˆ ko2 E‡ ‡ ka2 ÿ e i2qz Eÿ dz 2

…AII:13a†

d 2 Eÿ ˆ ko2 Eÿ ‡ ka2 eÿi2qz E‡ dz 2

…AII:13b†

and ÿ

In order to get rid of the exponential factors, one looks for solutions of (AII.13) in the form E‡ …z† ˆ ae i…l‡q†z ; Eÿ …z† ˆ be

i…lÿq†z

;

…AII:14a† …AII:14b†

and gets the following system of equations for amplitudes a and b: ‰…l ‡ q† 2 ÿ ko2 Ša ÿ ka2 b ˆ 0; 2

‰…l ÿ q† ÿ ko2 Šb ÿ ka2 a ˆ 0:

…AII:15a† …AII:15b†

2. Classroom Experiments with Chiral Liquid Crystals

63

Before analyzing solutions of these equations, let us brie¯y discuss polarization of light waves as a function of coe½cients a and b. It results from the de®nition (AII.12) that amplitudes of the electric ®eld are Ex …z† ˆ

E‡ …z† ‡ Eÿ …z† 2

…AII:16a†

Ey …z† ˆ

E‡ …z† ÿ Eÿ …z† 2i

…AII:16b†

and

or, explicitly, Ex ˆ Re‰Ex …z†eÿiot Š ˆ 12 a cos‰…l ‡ q†z ÿ otŠ ‡ 12 b cos‰…l ÿ q†z ÿ otŠ …AII:17a† and Ey ˆ Re‰Ey …z†eÿiot Š ˆ 12 a sin‰…l ‡ q†z ÿ otŠ ÿ 12 b sin‰…l ÿ q†z ÿ otŠ: …AII:17b† In Figure AII.1, we represented polarizations of the light wave for z ˆ 0 and for di¨erent values of a and b. This polarization is circular clockwise for a 0 0 and b ˆ 0 (wave E‡ ), circular anticlockwise for a ˆ 0 and b 0 0 (wave Eÿ ), linear for a=b ˆ G1, and elliptic everywhere else. In general, the polarization of the light wave can be represented as a superposition of two waves with opposite circular polarizations: E‡ with amplitude a and Eÿ with amplitude b. For z 0 0, the analysis of the polarization can be made in exactly the same way except for the fact that one has to take into account the phase shift of Dj ˆ 2qz, between the clockwise and anticlockwise circularly polarized wave components, which appears because of the di¨erence of 2q between the corre-

Figure AII.1. Polarization of the light wave for di¨erent values of the amplitudes a and b in (AII.17).

64

P. Pieranski Figure AII.2. In general, the eigenmodes of Maxwell equations in the cholesteric phase are waves with clockwise and anticlockwise elliptical polarizations. The axes of ellipses follow the rotation of the director in the cholesteric helix.

sponding wavevectors l ‡ q and l ÿ q. Let us note that this phase shift can by reduced to zero in the reference frame whose x-axis rotates with the cholesteric helix. Therefore, if the polarization is elliptical for z F 0, it stays elliptical for any z but the axes of the ellipse rotate in space with the cholesteric helix. This property was in fact expected because the fundamental property of eigenmodes is their invariance with respect to the symmetry of the medium. Now, let us come back to solutions of the system of (AII.15). From the condition Det ˆ 0, we get the dispersion relation linking the frequency o, through ka and ko , to the wave vector l: ‰…l ‡ q† 2 ÿ ko2 Š‰…l ÿ q† 2 ÿ ko2 Š ÿ ka4 ˆ 0:

…AII:18†

For a given o, this equation is of fourth order in l and has four solutions that form two pairs with opposite signs as is shown in Figure AII.3: r q 2 2 …AII:19† Gl2 ; Gl1 ˆ G …ko ‡ q † G 4ko2 q 2 ‡ ka4 : In the same ®gure are plotted ratios a=b or b=a corresponding to eigenmodes l1 and l2 . These ratios have been calculated using equations a=b ˆ ka2 =‰…l ‡ q† 2 ÿ ko2 Š

…AII:20a†

b=a ˆ ka2 =‰…l ÿ q† 2 ÿ ko2 Š:

…AII:20b†

and In the limit k g q, that is to say, when the pitch p is much larger than the light wavelength l, the two roots of (AII.18) are

2. Classroom Experiments with Chiral Liquid Crystals

65

Figure AII.3. Eigenmodes in the limit k g q where the cholesteric pitch p is much larger than the light wavelength.

l22 ; l12 A ko2 G ka2 or

p l 1 A e ? k ˆ no k

and

p l2 A ek k ˆ ne k:

…AII:21† …AII:22†

Using these solutions, one ®nds from (AII.20) that the modes l1 and l2 have orthogonal linear polarizations. These results con®rm the conclusions of Annex I. When the ratio k/q gets smaller, the polarizations of modes l1 and l2 become, respectively, elliptical clockwise and elliptical anticlockwise. The ratio a=b of model l2 decreases monotonically to zero with k in the limit k ! 0 and the polarization becomes perfectly circular anticlockwise. The behavior of model l1 is much more complex. When k is decreasing, ®rst its ellipticity (given by the deviation of a=b from ÿ1) increases. However, when k approaches a special value ku given by q …AII:23† ku ˆ p ; e? the ratio a=b tends to ÿ1 and the polarization becomes linear again. Between ku and kl given by q kl ˆ p ; …AII:24† ek

66

P. Pieranski

Figure AII.4. Eigenmodes in the case where the light wavelength l is close to (or much larger than) the pitch p.

the root l1 becomes imaginary which means that the mode cannot propagate in the cholesteric phase. The width of the forbidden interval ‰kl ; ku Š is a function of the anisotropy ea ˆ ek ÿ e? . This forbidden interval is known in X-ray crystallography as the Darwin band. For k ˆ kl , the polarization of the mode is linear, as for ku , but its direction is now orthogonal to the one at k ˆ ku . Finally, when k=q tends to zero, the polarization of the mode l1 becomes circular clockwise. In conclusion, in the limit k=q ! 0, the two eigenmodes have opposite circular polarizations and their wave vectors k1 ˆ l1 ‡ q and k2 ˆ l2 ÿ q are slightly di¨erent. This di¨erence between the wavevectors of modes with opposite circular polarizations is the reason for the rotatory power rˆ

k1 ÿ k2 2

…AII:25†

of the cholesteric phase whose pitch is much shorter that the light wavelength. Using (AII.19) it can be shown that rˆ

ea2 k 4 : 8q…q 2 ÿ ko2 †

…AII:26†

This expression is known as the de Vries formula of the rotatory power of the cholesteric phase.

3

From a Chiral Molecule to a Chiral Anisotropic Phase Hans-Georg Kuball and Tatiana HoÈfer

3.1

Introduction

The interest in chiral phases composed of chiral molecules or induced by a chiral dopant dissolved in an achiral phase has been mainly determined by their technological applications such as, e.g., in liquid crystal display technology. In order to ®nd suitable materials for these applications, many compounds with di¨erent structures have been synthesized to optimize the properties for these special technical purposes. Pioneering contributions to this ®eld have been published by Gerhard Heppke [1]±[8] and his group. Some of these results were summarized in a workshop in Berlin in 1995 [9]. Only for a few classes of compounds a qualitative correlation between the structure of a chiral compound or dopant and the chirality or helicity of the chiral liquid crystal phase has been developed; this was impeded by the absence of suitable and measurable structural quantities. It becomes apparent that no simple description for a structure/response relation can exist. Di¨erent mechanisms with di¨erent weighting factors seem to be responsible for the occurrence of the chirality or helicity of a phase. In general, this chiral induction is a process by which the molecular chirality is mapped onto a liquid crystal phase by inducing or changing the sign and size of a pitch of the cholesteric …N  † or smectic C  …SC  † phase, etc., or by producing other kinds of suprastructural chirality created by long-range orientational or positional order. This mapping starts from the molecular constitution of a chiral compound and takes into account the many possible conformers and their interaction with the molecules in the surroundings, the latter are chiral molecules of the same kind in a pure cholesteric phase or chiral guest or achiral host molecules in a guest/host phase. These interactions can produce chiral distortions of the molecules (or, resp., variations) in their chemical structures and a shift of enantiomeric ratios of the chiral conformers of host and guest molecules. To correlate the pitch or the helical twisting power (HTP), which is the reciprocal pitch normalized with a concentration unit, with the molecular structure, it is necessary to determine experimentally molecular properties which can be correlated to structural parameters of 67

68

H.-G. Kuball and T. HoÈfer

these molecules as well as to the HTP. However, it is not always well understood what kind of physical properties derivable from the molecular structure are responsible for the HTP. Without doubt, the absolute con®guration is very important because, under identical conditions, enantiomers induce a pitch of the same absolute value but of opposite sign. But do other usable properties exist? Furthermore, the question arises of whether the pitch or the reciprocal pitch is a chirality measurement or not [10], [11]. Therefore, an experimental program has been started to ®nd properties correlated to the structure of a molecule upon which a theoretical description of the chiral induction can be based [12]±[14]. Also in this chapter a route from the chiral molecule to the chiral phase will be developed, discussed, and illustrated by a few measurable phenomena. One can either focus on systems that consist only of chiral molecules or systems that are composed of chiral dopants (guest) and achiral molecules of the host phase. The phenomena that lead to the chiral induction are, at least in part, the same for both cases. In phases composed only of chiral molecules, the additional interactions between the chiral molecules have to be considered. To avoid this problem we decided to concentrate on guest±host systems where only small amounts of the dopant are dissolved. For the following discussion one fundamental assumption has to be made at the beginning, namely, that the induction process can be decomposed into two steps: ®rst, the intramolecular chirality transfer from a chiral element within the molecule to all other parts, the ``achiral parts,'' of the molecule; and second, the intermolecular chirality transfer from a molecular species (conformer) to the chiral phase. In this context, transfer basically means a transfer of the ``information chirality'' from one level of chirality to another.

3.2 3.2.1

Chirality of Molecular Systems Chirality of Molecules

A compound is characterized by its summation formula which shows the composition of chemical elements, its constitution which describes the connectivities of the elementsÐthe bonds between the elementsÐand its con®guration which speci®es the arrangement of the atoms in space. Considering the molecule as a rigid object as shown in the example of Figure 3.1, the application of symmetry operations of the second kind Sn , where n ˆ 1; 2; 3; . . . ; gives the multiplicity of the axis, allows us to decide whether a molecule is chiral or not, i.e., achiral. Chiral molecules which do not possess any symmetry element Sn exist in two enantiomeric forms, the enantiomers, which are mutually related as image and mirror image forms. The approach of molecular systems as rigid bodies is, in general, too simple

3. From a Chiral Molecule to a Chiral Anisotropic Phase

69

Figure 3.1. The S- and R-enantiomers of bromochloro¯uoromethane [15].

because in reality they possess dynamic structures with lifetimes in very different time domains. One type of molecular chirality which is of interest is obtained by the rotations of groups about s bonds allowing the formation of more or less stable chiral formsÐso-called conformers or rotamers (Figure 3.2). Variations of the geometrical degrees of freedom in the time domains of vibrations are not of interest here. To handle problems connected to chirality it makes sense to di¨erentiate between the con®gurational chirality in Figure 3.1, where chemical bonds have to be broken to transform one enantiomer into the other, and the conformational chirality in Figure 3.2, where a rotation, e.g., about a s bond, transforms one enantiomer into the other. Whether these conformers (rotamers) are stable or not depends on temperature, pressure, etc. As long as the conformers are thermodynamically stable, each of them will be taken as an individual species. In general, racemic±achiral compounds, i.e., mixtures of the enantiomers having R and S or P and M con®gurations, can be decomposed into the two chiral forms [16]. Compounds which are a mixture of their enantiomers are often called racemic achiral not only in the case of con®gurational but also in the case of conformational chirality. If one of the enantiomeric forms dominates over the other, the material possesses an enantiomeric excess (ee) of one form. Conditions may exist where not even one of the chiral conformers of a compound is thermodynamically stable, then only one conformationÐthe free rotation in the case of 1,2-dichloroethaneÐ exists. To di¨erentiate between chiral forms does not make sense in these cases. The symmetry operation has to be applied to the ``mean'' con®guration of the conformation.

Figure 3.2. The two enantiomeric forms of a biphenyl and a 1,2-dichloroethane.

70

H.-G. Kuball and T. HoÈfer

Often symmetry operations cannot be used in a simple way to classify chiral forms because, e.g., the molecule consists of a number of conformations. Therefore, independent of the symmetry considerations, a chemical approach to describe chiral molecules has been introduced by the use of structural elements such as chiral centers, chiral axis, and chiral planes. Examples for a chiral center are the asymmetric carbon atom, i.e., a carbon atom with four di¨erent substituents or the asymmetric nitrogen atom where a free electron pair can be one of the four di¨erent substituents. A chiral axis exists with a biphenyl (Figure 3.2) and chiral planes are found with cyclophane structures [17]. Chiral elements were introduced originally to classify the absolute con®guration of molecules within the R, S nomenclature [16]. In cases where the molecules are chiral as a whole, so-called inherent dissymmetric molecules, special names have often been introduced: atropisomers, i.e., molecules with hindered rotation about a s bond, ``GelaÈnder'' helical molecules [18], calixarenes, cyclophanes [17], dendrimers [19], and others [20]. With the development of sector or helicity rules for the determination of the absolute con®guration with circular dichroism spectroscopy it has become absolutely necessary to divide molecules into classes. Every class then possesses its own rule [21]±[23]. For similar reasons at least three structural criteria for classes of dopant (guest) molecules have to be considered for deriving rules for the development of structure/response relations for the HTP [12]±[14]: Class A: Molecules with one, two, three, or more chiral centers/elements far away from each other. Each chiral center/element acts as an independent chiral unit. Class B: Molecules with two or more chiral centers/elements which are close together. All these centers together act like one chiral unit. Class C: Inherently dissymmetric molecules.

3.2.2

Chirality of Phases

Chiral phases can be formed from achiral, racemic achiral, or chiral atoms/ molecules. In a gas, a liquid, a liquid crystalline, and a crystalline phaseÐas long as external forces are absentÐan isotropic orientational distribution of chiral molecules can exist. Such phases are often named as isotropic dissymmetric phases. Chiral, i.e., pseudoscalar, e¨ects of the phase can then be traced back to properties of single molecules. In liquids, short-range orientational and positional order in a ``solvent cage'' may contribute to a chirality measurement, i.e., a measurement of a pseudoscalar quantity. Pretransitional e¨ects may contribute to the chirality of a phase, this has been shown experimentally [24] and explained theoretically as a ¯uctuation phenomenon [25], [26]. In liquid crystalline phases or crystals, long-range interaction leads to long-range orientational and long-range positional order of

3. From a Chiral Molecule to a Chiral Anisotropic Phase

71

the molecules by which a suprastructural chirality can be obtained. Anisotropic structures with chiral elements may appear, such as, e.g., a screw axis in cholesteric or smectic C  phases or more than one screw axis in blue phases. More complex structures are possible by di¨erent kinds of longrange orientational and positional order. In general, chirality in phases is no longer a phenomenon of the molecules themselves, it is given by the chiral subsystem ``molecule'' and additionally by chiral long-range orientational and positional order. Here, chiral molecules are the origin of the suprastructural chirality. It should be kept in mind that chiral phases without suprastructural chiralityÐthe A  phase [27]Ðexist as do, in contrast, phases with suprastructural chirality where achiral or racemic achiral molecules form a suitable long-range positional and/or orientational order [28], [29].

3.2.3

Four Levels of Chirality

Objects like atoms, molecules, and ensembles of molecules are chiral. Thus, properties connected to the chirality of an object may have very di¨erent origins. Therefore, it makes sense to introduce a concept in which four levels of chirality exist. The ®rst level of chirality is the chirality of the atoms [30] caused by weak interaction which is of no interest for the discussion of liquid crystal properties. The second level is the chirality of molecules, while the third level is derived from the ordering of atoms, ions, or molecules in isotropic or anisotropic phases by long-range positional and long-range orientational order. The fourth level of chirality is the form of a macroscopic object which can be, e.g., an enantiomorphic crystalline form (habitus of the crystal).

3.2.4

Chirality Transfer Between the Four Levels of Chirality

With a four-level concept of chirality, the question of how chirality can be transferred from one level to an adjacent level is of fundamental interest. In particular, the mapping of the chirality of a molecule (second level) onto the liquid crystal phase (third level) or vice versaÐthe intermolecular chirality transferÐwill be discussed within the scope of this chapter. But what does chirality transfer really mean? The questions of how much chirality a molecule possesses or how much chirality can be transferred to an adjacent level are not allowed in the sense of transport of an amount of ``chirality'' from one level to another. Chirality is an information which can be transferred from one level to the other without being lost in the level from which it is originated. There are two possible statements [10]: (1) The information about the absolute con®guration in one level is transferred to and stored by the absolute con®guration of the adjacent level.

72

H.-G. Kuball and T. HoÈfer

(2) The results of some of the chirality measurements of one level are correlated to the results of the chirality measurements of the adjacent level. The correlated chirality measurements do not have to be of the same type but not all possible chirality measurements can be taken for a correlation. For the ®rst point, a convention for the absolute con®guration of each level is needed. For molecules (second level) the Cahn±Ingold±Prelog nomenclature [15] R or S and P or M is available whereas, for example, the helicity of a cholesteric or a smectic C  phase (third level) can be given by P or M. Such a simple characterization is not possible for all chiral liquid crystal phases. If we consider that a chiral dopant (the second level), the object, with a con®guration R, for example, induces a cholesteric phase of helicity P, the information ``absolute con®guration R'' of the second level is transferred to the helicity P of the third level. If the chirality of a number of compounds and also their e¨ect of mapping onto the phase are correlated as demonstrated schematically in the following diagram, a rule for the determination of the absolute con®guration of an object is obtained: Second level

abs. con®g. of object 1

$

l Third level

abs. con®g. of the ind. phase 1

abs. con®g. of object 2

$

l $

abs. con®g. of the ind. phase 2

abs. con®g. of object 3

$

l $

abs. con®g. of the ind. phase 3

........ l

$ ........

In the literature, the mapping of the molecular chirality onto a liquid crystal phase has been interpreted, also by our group, as an e¨ect of ampli®cation of the chirality of a molecule. This language should be avoided under the consideration of the second statement. The chirality measurements for the adjacent levels can be of the same or of a di¨erent kind. The Cotton e¨ect of the dopant (second level) can be compared to the Cotton e¨ect of the induced cholesteric phase (third level) which is, in general, larger by orders of magnitude. The sign of the Cotton e¨ects of the molecule belonging to di¨erent absorption bands depends on the nature of the transition and the surroundings of the chromophore, the Cotton e¨ects of the phase depend on the suprastructural chirality around the molecule and, therefore can only be compared and correlated in special cases. In all cases, however, it is not an enhancement of the chirality of a molecule. But it is always an enhancement of values of chirality measurements of di¨erent origins. The detection of a chiral compound by its chiral induction of a cholesteric phase is more sensitive than any other method. The e¨ect can be found without a concentration threshold in very dilute solution where one solute molecule in¯uences a hundred or more solvent molecules. In this case there is

3. From a Chiral Molecule to a Chiral Anisotropic Phase

73

no chirality measurement corresponding to the HTP with the single molecules. Here, another kind of chirality measurement has to be taken instead. A chiral interaction potential from a ``chiral surface'' may be a useful quantity [31]±[37] or, in suitably chosen examples, even the circular dichroism (CD) or anisotropic circular dichroism (ACD) of the corresponding compound [38], [39]. A correct notion and the only notion that makes sense for this enhancement is that of an ``ampli®cation of a value of a chirality measurement'' because then quantities of the same kind can be compared, e.g., the optical rotation or the circular dichroism. Up to now, the chirality transfer from the level of molecules (second level) to the phase (third level) has been taken into consideration. An example of the transfer of chirality from the liquid crystal phase (third level) to the molecule (second level) is the liquid crystal induced CD (LCICD) [40], [41].

3.3

Phenomenological Equations and Basic Experimental Findings

3.3.1

De®nition of the Helical Twisting Power and Phenomenological Dependences

The helical twisting power (HTP) has to be de®ned in a generalized form [12]±[14] by ( )   ÿ1  1 qpÿ1 qp ÿ ; …3:1† …HTP†e ˆ qxe xe ˆ0 qxey x y ˆ0 2 e

y

where e denotes the enantiomer of e and xe , xey are the mole fractions of e and ey , respectively. The di¨erential quotient of the reciprocal pitch pÿ1 with respect to the mole fraction xe is not a symmetry adapted function for a chiral guest in a chiral liquid crystal phase. By a linear combination of both derivatives according to (3.1) and (3.2) a pseudoscalar and a scalar quantity, i.e., two symmetry adapted quantities,1 are created. The achiral helical twisting power (AHTP) is then ( )   ÿ1  1 qpÿ1 qp ‡ : …3:2† …AHTP†e ˆ qxe xe ˆ0 qxey x y ˆ0 2 e

The AHTP is di¨erent from zero not only for a chiral compound but also for an achiral compound dissolved in a chiral phase or a compensated nematic

1 This procedure corresponds to the construction of the CD and the mean absorption of an isotropic dissymmetric solution from the absorption coe½cients of left and right circularly polarized light [42].

74

H.-G. Kuball and T. HoÈfer

phase. If the concentration of a chiral compound dissolved in its own induced chiral phase is increased there may be a change of the HTP on account of the nonideal behavior of the mixed phase. Here, the AHTP is always zero because there is no diastereomeric interaction and the derivative of the reciprocal pitch ful®lls the condition of a chirality observation. This is consistent with  ÿ1   ÿ1  qp qp ˆÿ ; …3:3† qxe xe ˆ0 qxey x y ˆ0 e

which is also ful®lled for a chiral guest in an achiral host phase. For a nematic host phase (3.3) holds and the HTP is given by  ÿ1  qp ˆ ÿ…HTP†ey : …3:4† …HTP†e ˆ qxe xe ˆ0 For dilute solutions the e¨ect is a linear function of the guest concentration. If more than one chiral guest, e.g., di¨erent chiral compounds, contribute to the chiral induction for very dilute solutions it has been found that the HTP can be described in a good approximation as X xi …HTP†i ; …3:5† HTP ˆ i

where xi is the mole fraction and (HTP)i is the HTP of the species i. From the thermodynamic point of view the guest±host systems have to be treated as mixed phases and the HTP has to be correlated to a partial molar quantity. From this point of view the conclusion that the HTP depends also on the properties of the achiral molecule, the host systems in a guest±host system, for example, is a natural consequence. The solvent dependence is shown, as an example, with the aminoanthraquinones R-3 and R-12b measured in the four components of ZLI-1695 (Merck) in the region of their nematic states (Figure 3.3). One class of compounds, where an intramolecular and an intermolecular chirality transfer and, second, one class of molecules, where only an intermolecular chirality transfer can be expected, will be discussed below. The aminoanthraquinones were chosen as model compounds (Figure 3.4) because they consist of a large achiral rigid part and chiral substituents of a similar kind positioned in di¨erent locations on the anthraquinone skeleton. The anthraquinone skeleton as a large rigid part determines mainly the order and accordingly the chiral elements are ``carried in di¨erent orientations'' into the phase. A large variation of the HTP has been found for these compounds. The second class chosen consists of inherently dissymmetric compounds (Figure 3.5) which possess only an intermolecular chirality transfer. Here, a well-known and often chosen class of compounds, the binaphthyls, is selected. The HTP of all compounds was determined with the help of the Cano method [2].

3. From a Chiral Molecule to a Chiral Anisotropic Phase

75

Figure 3.3. Host dependence of (a) R-12b and (b) R-3 in ZLI-1695 (9) and its four components (R ˆ C2 H5 (y), R ˆ C3 H7 (i), R ˆ C4 H9 (h), R ˆ C7 H15 (G)) in the temperature region of their nematic phases.

Figure 3.4. HTP as a function of temperature in ZLI-1695 (Merck) of the aminoanthraquinones R-1a, R-3, R-7, R-12a‡ , R-14, R-15, R-34, and R-36. The su½x ‡ indicates that the enantiomer has been measured.

76

H.-G. Kuball and T. HoÈfer

Figure 3.5. The helical twisting power as a function of the reduced temperature in ZLI-1695 (Merck) of R-B1 (y), R-B3 (‡), R-B4 (r), R-B5 (i), R-B6 (), and R-B7 (h) (the experimental error is given by the size of the symbols for the measured points).

3.3.2

Description of Order

The anisotropy of liquid crystal properties is determined by the long-range positional and orientational order of the molecules, which can be described by an orientational distribution function and a distribution function for the long-range positional order f …W; r† which can be a periodic or a very complex function. E¨ects introduced by ¯uctuations will not be discussed here. For the description of a number of phenomena it is su½cient to introduceÐ especially localÐorder parameters which can be given in the form of coe½cients of Legendre polynomials. For many optical properties it is more convenient to introduce tensorial parameters which are given with respect to the rank of the tensors describing a property of the anisotropic phase Mijkl... and molecules Xijkl... . For phases with an isotropic long-range positional order, a set of orientational distribution coe½cients gijklmn... can be introduced for each species of the phase … 1 f …W†aim …W†ajn …W†akr …W†als …W† . . . dW: …3:6† gijklmnrs... ˆ N W 1 …a; b; g† are the Eulerian angles and aij …a; b; g† are the elements of the transformation matrix from the space-®xed to the molecule-®xed coordinate system. The properties of the phase are then given by a sum of products of

3. From a Chiral Molecule to a Chiral Anisotropic Phase

77

one factor which describes the order and another factor which belongs to a molecular property X gijklmnrs... Xijkl... : …3:7† Mmnrs... ˆ 1; m; n;...

The advantage of this, sometimes cumbersome description, is the fact that the transformation properties of the macroscopic and microscopic coordinates can be seen immediately. Because the anisotropy of most of the phases is su½ciently described by tensors of second rank for the microscopic as well as the macroscopic properties, only the orientational distribution coe½cients gijkl are mentioned here. As an example for the description of order by these orientational distribution coe½cients, the order of a cholesteric phase will be brie¯y discussed. Four di¨erent order parameters are needed for a molecule of point symmetry group D2 and local symmetry D2 for the cholesteric phase.2 The stars as a su½x indicate that gij33 is given in its system of principal axes. In general, the convention    V g2233 V g1133 g3333

…3:8†

is used below unless otherwise mentioned. Instead of the coordinates of the order tensor gij33 often the Saupe's ``coordinate-free'' order parameters are used as a short-cut description. The four order parameters S  , D  , A  , and B  then required for a cholesteric phase if the microscopic biaxiality of the phase is taken into account are  ÿ 1† ˆ h12…3 cos 2 b ÿ 1†i; S  ˆ 12…3g3333 p p 3  3   …g2233 ÿ g1133 † ˆ ÿ hsin 2 b cos 2gi; D ˆ 2 2 p p 3  3   …g3322 ÿ g3311 † ˆ ÿ hsin 2 b cos 2ai; A ˆ 2 2

…3:9† …3:10† …3:11†

and     ÿ g1122 ÿ g2211 ‡ g2222 † B  ˆ 12 …g1111

ˆ h12…1 ‡ cos 2 b† cos 2a cos 2g ÿ cos b sin 2a sin 2gi:

…3:12†

S  and D  refer to the optical axis of the uniaxial phase which is chosen as the space-®xed x30 -axis and to the molecule-®xed xi coordinate system. Whereas S  characterizes the order of the orientation of the molecule-®xed x3 -axis with respect to the optical axis, the parameter D  is a measure for the deviation from a rotationally symmetrical distribution of a molecules about

2 Fifteen parameters are needed for the long-range orientational order for a C  phase with local C2 symmetry of the phase and C1 point symmetry group for the molecules [11] which is too cumbersome for a discussion of this case as an example.

78

H.-G. Kuball and T. HoÈfer

Figure 3.6. The order triangle is a graphical representation of all possible values of the order parameters S and D. The lines L1 to L4 are characteristic for special sit      ˆ g2233 , L2 : g3333 ˆ g2233 , L3 : g2233 ˆ 13, L4 : g1133 ˆ 0. uations of the order: L1 : g1133   points for the long-range oriThe points Pi …S ; D † for i ˆ 1 to 4 are outstanding p p entational distribution: P1 …0; 0†, P2 …1; 0†, P3 …14 ; 3=4†, P4 …12 ; 3=6†, P5 ˆ P…S  ; D  †.

its x3 -axis. The principal axes of gij33 are ®xed by symmetry for molecules with a point symmetry group di¨erent from C1 , C2 , Ci , Cs , and C2h [43]. For all other cases the orientations of the xi -axes with respect to the molecular skeleton have to be determined experimentally (determination of all elements of gij33 ); the orientations of the xi -axes vary, in general, with temperature. The order triangle (Figure 3.6) describes the area of existence of the order parameters [43], the knowledge of which is often very important for an experimental analysis. Only the order parameters …S  ; D  †, e.g., P5 in Figure 3.6, characterize the order in a way which is suitable to compare the order of di¨erent compounds. When the order parameters are given with respect to any other coordinate system, e.g., to the principal axes of the transition moment tensor (transition moment direction) in two di¨erent compounds, then the comparison of the order of these molecules does not make any sense. This can be demonstrated by the hexagon within the order triangle (Figure 3.6) which gives the region of ``apparent order parameters'' S and D for a pair of S  , D  values indicated with P5 (0.48, 0.18) if the reference axes, characterizing the transition moment tensor, for example, are rotated in all possible orientations with respect to the principal axes of the order tensor. The ``trefoil'' describes the nonstable region of a nematic phase within the scope of the

3. From a Chiral Molecule to a Chiral Anisotropic Phase

79

Figure 3.7. The order parameters for R-B1 (y), R-B3 (‡), R-B4 (r), R-B5 (i), R-B6 (), and R-B7 (h) in ZLI-1695 (Merck) determined by 2 H NMR spectroscopy. The solid lines represent D  ˆ f …S  ; d† [47], where d is the temperature-independent ratio of mean ®eld potential (from below: d ˆ 0:5, 0.55, 0.85, 0.95, 1.0, 1.2).

mean ®eld theory [44]. The S  and D  values in the triangle P1 P3 P4 yield a positive degree of anisotropy for a transition moment parallel to the x2 -axis. A  and B  describe the anisotropy of the internal structure of the phase [45]. The order parameter A  is as large as S  and B  is small, like D  . A  and B  are not available from a simple experiment. They are not needed for optical e¨ects like circular dichroism (CD) or optical rotatory dispersion (ORD) because of the symmetry of the molecular tensor which is responsible for CD and ORD. Their in¯uence on chiral e¨ects, like the HTP, is still unknown. The order parameters S  , D  and the principal axes of the order tensor can be determined by numerous methods. S  , D  for the 1,1 0 -binaphthyls (Figure 3.5) are obtained from 2 H NMR and 13 C NMR spectroscopy and con®rmed by the polarized ultraviolet (UV) spectroscopy (Figure 3.7). One of the principal axes of the order tensor is parallel to the C2 symmetry axis. The two remaining principal axes are rotated around the C2 -axis by a few degrees [46]. The temperature dependence of the orientation of these axes with respect to the molecular frame between 25  C and 70  C is less than 2 . For R-B4, R-B5, and R-B6 the x3 -axis is approximately parallel to the naphthyl±naphthyl bond, whereas x2 is parallel to the C2 symmetry axis. The orientation axis of R-B7 is switched [43] in comparison to R-B4, R-B5, and R-B6. Then x3 is parallel to the C2 symmetry axis, whereas x2 is approximately parallel to the naphthyl±naphthyl bond. For R-B1 and R-B3 the axes can be obtained by exchanging the axes of R-B4 …x1 ; x2 ; x3 † to …x2 ; x1 ; x3 † and …x3 ; x1 ; x2 †, respectively. For R-B3 there is an unsolved

80

H.-G. Kuball and T. HoÈfer

problem with the assignment of the axes resulting from the analysis of the 2 H NMR and the polarized UV spectra [46].

3.3.3

Intramolecular Chirality Transfer

Considering chiral structures and the corresponding chirality measurements, the question arises whether there are parts of a molecule which are more responsible for the result of these chirality measurements than others. Cahn, Ingold, and Prelog [15] introduced for nomenclature reasons chiral elements as the geometrical origin of chirality. In attempts to trace the results of chirality measurements back to the structure [11], [10] it becomes obvious that these chiral elements are of special importance. Parts of the molecules without chiral elements, the ``achiral parts,'' can be responsible for very different but, in general, smaller experimental values of the chirality measurement. This means that the same chiral element as shown in Figure 3.8 induces a very di¨erent CD if di¨erently bound or oriented with respect to the achiral parts. Here a di¨erent intramolecular chirality transfer has been

Figure 3.8. HTP as a function of temperature in ZLI-1695 (Merck) and the CD spectra of R-1a (





), R-2‡ (ÿ
ÿ
ÿ), and R-52 (ÐÐÐ) in n-heptane at 20  C which are approximately equal to those in ZLI-1695 (Merck) at 80  C. The su½x ‡ indicates that the enantiomer has been measured.

3. From a Chiral Molecule to a Chiral Anisotropic Phase

81

Figure 3.9. (a) Exciton coupling of two transitions localized in di¨erent parts of a molecule. The exciton splitting 2Vij is a consequence of a dipole±dipole interaction. (b) The CD of the exciton transitions and the result of a superposition to a CD couplet. DDe is the amplitude of the couplet [48].

introduced which makes sense with the following de®nition.3 The ``chiral part'' possess a chiral element and is, therefore, the geometrical origin of the chirality of the molecule. The ``achiral part'' is the part of the molecule which does not have any chiral element. This part is achiral or racemic achiral in the absence of the ``chiral part.'' With this concept of a formal decomposition, it should always be kept in mind that the molecular regions taken as the ``chiral'' and the ``achiral'' parts, respectively, can only be qualitatively de®ned in principle. Furthermore, the concept of transferring chirality from a ``chiral region'' and its near surroundingsÐthe chiral part of the moleculeÐto the residue of the moleculeÐthe ``achiral part''Ðhas to be proven. Within this concept, successfully applied in the theory of optical activity, chirality can be transferred from a chiral element to the residue of the molecule by which new chiral structures are produced. A distortion of the electronic structure or an excess of a chiral conformation can be induced in the ``achiral part'' of the molecule. CD spectroscopy can serve as a tool to determine this transfer. For example, the CD of a chromophore decreases if the distance between the chromophore and the chiral center increases. This can be demonstrated by comparing the CD of aminoanthraquinones R-1a and R-52 in Figure 3.8. For substituted aminoanthraquinones the transfer

3 In general, only a conformer as a whole is a chiral or achiral species in terms of the de®nition of Lord Kelvin. In our concept we require the decomposition of a molecule into two parts. One of these parts is chiral and the other part is ``achiral'' in the sense of the de®nition of Lord Kelvin. Since there is no special notation available for these parts, we call them the ``chiral and achiral parts'' of the molecule because there is no better name for the part of a molecule where a chiral element is located and that part where there is no chiral element.

82

H.-G. Kuball and T. HoÈfer

of the ``information chirality'' from one and the same chiral ligand to the anthraquinone skeleton depends on the position of substitution of the ligand on the skeleton (R-1a, and R-2‡ in Figure 3.8 [14]). That there is no transfer between groups far away from each other can be shown by comparison of the CD spectra of R-11a‡ , R-12a‡ , and R-53. The chirality information from the chiral center is transferred to the aminoanthraquinone skeleton, this is demonstrated by the CD of the charge transfer transition of the anthraquinones. Furthermore, it has been found that the contributions of the groups in 1,4- or 1,5-positions to the CD of the charge transfer transition are additive as expected from the plain chirality function for C2v inherent symmetric chromophores [11], [14]. But there is no transfer from the chiral center in the 1-position to the region of the naphthyl group in R-53 because no exciton couplet [48] can found in the spectral region around 44,000 cmÿ1 in contrast to the CD spectrum of R-11a‡ (Figure 3.10). This means that there is no evidence for an excess of a chiral conformer in R-53 induced by the substituent in the 4-position and the anthraquinone skeleton analogously to that in the 1-position of R-11a‡ .

Figure 3.10. The CD of R-11a‡ (





), R-12a‡ (ÿ
ÿ
ÿ), and R-53 (ÐÐÐ) in nheptane at T ˆ 20  C. The su½x ‡ indicates that the enantiomer has been measured.

3. From a Chiral Molecule to a Chiral Anisotropic Phase

83

The presence of conformers can be shown by the comparison of the amplitude of the exciton couplet of the CD spectra of the naphthylethylaminoanthraquinones R-5‡ , R-6‡ , R-11a‡ , and R-16‡ in the ZLI-1695 solution. The temperature dependence of DDe…T† can be used [49] to determine chiral conformers by X xi Dei ; …3:13† De ˆ i

where the Dei are the CD spectra of the chiral conformers. Taking (3.5) and so the assumption that every conformer of a compound contributes di¨erently, i.e., with (HTP)i for the ith conformer in accord with its concentration xi to the HTP, then the temperature dependence of the CD spectrum and the HTP can be compared by the temperature dependence of the mole fractions xi …T†. A linear relation between the HTP and the amplitude of the exciton couplet results: HTP…T† ˆ K1 DDe…T† ‡ K2 ;

…3:14†

where K1 and K2 are functions of the (HTP)i and of Dei of the involved conformers [50]. The linear dependences shown in Figure 3.11 give evidence for the presence of conformers and con®rm that every conformer of a com-

Figure 3.11. (a) DDe…T† measured in n-heptane [51] and (b) HTP…T† measured in ZLI-1695 as a function DDe…T† measured in n-heptane for the naphthylethylaminoanthraquinones R-5‡ (), R-6‡ (9), R-11a‡ (v), and R-16‡ (e) [50], [12]. The su½x ‡ indicates that the enantiomer has been measured.

84

H.-G. Kuball and T. HoÈfer Figure 3.12. The orientation axis …x3 † is de®ned as the principal axis of the order tensor to which the largest eigenvalue belongs. The axis parallel to the longest extension of the molecule is often taken as an approximation for the orientation axis.

pound contributes di¨erently to the HTP, i.e., with (HTP)i for the ith conformer, in accord with its concentration xi .

3.3.4

Intermolecular Chirality Transfer

As discussed, every stable conformer of a compound acts as a separate species. Nonrigid molecules have to be discussed with respect to the lifetimes of their di¨erent conformations. Furthermore, the orientation of the orientation axis, which can be determined by 2 H NMR and partly by 13 C NMR spectroscopy [52], is important for the size and sign of the HTP [13], [14]. A variation of the orientation of the orientation axis x3 with respect to the molecular skeleton can be achieved by di¨erent substituents R…i† or R… j† as demonstrated in Figure 3.12. A variation of the temperature of the host system can also lead to a change of the orientation axis for molecules of low symmetry, i.e., with the molecules possessing a point symmetry group C1 , C2 , Ci , Cs , and C2h . For compounds of class A (see Section 3.2.1) the HTP is approximately the sum of contributions from di¨erent chiral centers. Two identical chiral centers only double the e¨ect if their orientations with respect to the orientation axis are the same. Otherwise, they contribute with di¨erent values to the HTP. This e¨ect can be seen immediately by comparing the HTP values of R-12a‡ and R-7 (Figure 3.13) where even a change in sign has been found with the same substituent. Here, in contrast to the mono-substituted R-12a‡ , the bis-substituted compound R-7 possesses an orientation axis which is rotated from the C2 -axis in R-7 to the ``long axis'' in R-12a‡ [14]. Comparison of the result for R-34, where the orientation axis is rotated by the achiral substituent in the 4-position approximately parallel to the long axis of the anthraquinone skeleton, with the result for R-7 shows (Figure 3.13) that the e¨ect for the chiral ligand in R-34 is approximately half of the value for R-7.

3. From a Chiral Molecule to a Chiral Anisotropic Phase

85

Figure 3.13. The orientation of the orientation axis x3 within the compounds R-7, R-12a‡ , R-15, and R-34 and the HTP in ZLI-1695 at T ˆ 50  C [52]. The su½x ‡ indicates that the enantiomer has been measured.

These results allow the statement to be made that chiral substituents have the same contribution to the HTP if the molecule has the same orientation axis and the substituent has the same orientation4 with respect to the orientation axis. The same substituents in molecules with di¨erent orientation axis have di¨erent HTP values even if S  and D  are the same. Group contributions have been evaluated for the aminoanthraquinones with di¨erent substitution pattern (Table 3.1). As can be seen in Table 3.1 the HTP is dependent on the number of chiral centers/elements per molecule for guest molecules of class A [13]. The numerical contribution of a group is also determined by the orientation of the group with respect to the orientation axis. Furthermore, the additivity of the e¨ects of chiral centers requires that the chiral centers are far away from each other. This result [12]±[14] is not in contradiction to earlier ®ndings [53] that the HTP does not depend on the number of chiral centers. Here, com4 The notion ``the orientation of the chiral group with respect to the orientation axis of the guest'' is qualitative at the moment. But, possibly, the introduction of the tensor equations in Section 3.4 will allow us to use the principal axes of the chirality tensor Cij to describe this orientation.

86

H.-G. Kuball and T. HoÈfer

Table 3.1. Contribution of the R-arylethylamino or R-cyclohexylethylamino substituents to the HTP of aminoanthraquinones (HTP per substituent/mmÿ1 ). Phenylethylamino

1-Naphthalen-1ylethylamino

Cyclohexylethylamino

1

50

31

10

1.5

57

38

9

1.4

37

13

ÿ3

1.8

31

9

ÿ8

Substitution pattern

pounds of class B have been analyzed where the chiral centers are not far away from each other, i.e., the chiral centers act as one chiral unit.

3.3.5

The In¯uence on the Molecular Extension

For the atropisomers of the biaryl type it has been found that their rigid and inherent dissymmetric form causes the large chiral induction. For these compounds (Compounds of class C, Section 3.2.1) a successful qualitative structure/response relation has been derived experimentally [54]±[57]. Furthermore, a successful theoretical prediction of the HTP is possible by the so-called ``surface model'' where the surface of the atropisomers, on the one hand, determines the order parameter and, on the other hand, the HTP in a good approximation [31]±[37]. Therefore, it make sense to look for further direct measurable e¨ects which con®rm such a conclusion. With the TADDOLs it can be shown (Figure 3.14) that the HTP increases approximately proportional to the extension of the aromatic substituents phenyl:1naphthyl:2-naphthyl:phenanthryl with a ratio 1:2:2:2:3 [58], [59]. R; R-T17 does not ®t into the scheme possibly because of the unrestricted rotation of the phenyl groups. Furthermore, from the experimental data (Figure 3.5) of the binaphthyls R-B4 and R-B5 the conclusion can be drawn that the O±C±O plane together with each of the two naphthyl planes (Figure 3.15) introduces a new active chiral element into the molecule. The dihedral angles of this new plane with the two naphthyl planes change as a function of the dihedral angle of the naphthyl±naphthyl bond.

3.3.6

Helix Inversion

The origin of the sign inversion of the reciprocal pitch of a compound without changing pseudoscalar parameters seems to be one of the most fascinating questions to be answered. But such a phenomenon of sign inversion as a

3. From a Chiral Molecule to a Chiral Anisotropic Phase

87

Figure 3.14. The HTP of the TADDOLs R; R-T2, R; R-T3, R; R-T7, R; R-T8, and R; R-T17 as a function of temperature in ZLI-1695 (Merck).

function of variables, which are not related to chirality, as temperature, concentration, or the composition of the host phase, etc., is well known for other chirality measurements like CD. At present, there are four mechanisms known and proven for the helix inversion: (1) There are conformers in an equilibrium which contribute according to (3.5) to the HTP of the phase with (HTP)i values of di¨erent sign [6]. A shift of the equilibrium with temperature can then lead to a sign change. (2) In a molecule of class A with two chiral centers, where each center induces a twist of di¨erent sign and size and each contribution has a di¨erent temperature-dependence [60], [61] a change of sign may result from the opposite e¨ect of the two centers by temperature variation. (3) The orientation of the orientation axis or, more generally, the principle axes of the order tensor of the guest molecule with respect to the molec-

Figure 3.15. A rotation about the dihedral angle of the naphthyl±naphthyl bond of the plane de®ned by the atoms O±CR2 ±O introduces a new chiral element into the molecule.

88

H.-G. Kuball and T. HoÈfer

ular skeleton can change, e.g., with temperature. In principle, this can cause a sign change of the HTP [13], [14]. (4) A theory of Lin-Liu et al. [62], [63] and Zakhlevnykh [64], which requires at least two independent pseudoscalar potential parameters for one and the same molecule, predicts a helix inversion for a rigid molecule when each parameter introduces a contribution of di¨erent sign and with a di¨erent temperature dependence. The helix inversion which corresponds to the mechanisms (1) and (2) is not unexpected, as these e¨ects are known from CD spectroscopy. The mechanisms (3) and (4) are unknown in CD spectroscopy. These e¨ects are caused by the fact that the HTP is a result of the interaction of an oriented guest molecule with its surroundings built up of oriented host molecules, i.e., an e¨ect of an anisotropic interaction. The e¨ect of (3) has its analogy with the CD spectra of oriented molecules (ACD spectroscopy [65]) because ACD spectra can change their sign as a function of temperature and if measured with a light beam propagating along di¨erent directions through the molecule dissolved, e.g., in a uniaxial phase.

3.4 3.4.1

Theoretical Models and Their Application Sector or Helicity Rules for the HTP as a Chirality Measurement

The HTP and the reciprocal pitch pÿ1 for the chiral induction are values of chirality measurements. Hence, the pseudoscalar quantities HTP or pÿ1 change sign analogously to the CD …De† when the system is composed of the enantiomers as long as equal experimental conditions exist. This means that the quantities, HTP, pÿ1 , and De are correlated to the absolute con®guration. The question arises whether a relation between the sign of the HTP and De of a compound also exists. The current leading opinion is that no relation can be obtained because of the very di¨erent physical mechanisms which lead to De or to the HTP. The optical information De stems from the interaction between light and the molecules which is described by quantities like the electric dipole, magnetic dipole, and, for the anisotropic system, also electric quadrupole transition moments. The HTP is determined by static interactions between the guest and host molecules or, more accurately, between the interaction between that part of the host molecule which is responsible for intermolecular chirality transfer. But this is not the complete argumentation because both e¨ects, CD and HTP, can be traced back to the same origin. Both types of interaction transport the same information as long as the phase ``sees'' the geometrical region of the molecule which is responsible for the CD. Thus, the question arises whether the phase can be a

3. From a Chiral Molecule to a Chiral Anisotropic Phase

89

Figure 3.16. Sector rule for the helical twisting power of a chiral guest: HTP A A x
y
z where A is a complex function of the position of the perturbing atom.

``label'' for a group of atoms within the molecule which are a chromophore for the CD or induce CD in another chromophore of the molecule. Because the chromophore is then the label for the same group of atoms for the CD spectroscopy, sector, and helicity rules [21]±[23], [66] like those for CD should also exist for the HTP [67]. This can be shown by a simple model used to develop sector rules for the determination of the absolute con®guration with the help of CD spectroscopy. In Figure 3.16 a model molecule is presented which consists of a plane with D2h symmetry which is achiral and a ®xed perturber which introduces chirality into the molecule. This model molecule possesses a distinct HTP and a distinct De. Moving the perturbing atom from the point P…ÿx; y; z† along a straight line parallel the x-direction, e.g., to the point P…x; y; z† (Figure 3.16), the model molecule is transformed into its enantiomer. When the point is at P…0; x; y†, the system is achiral because the perturbing atom is in one of the mirror planes of the skeleton and thus the HTP and De are identically by zero. If there is no chiral zero for the perturber within an octant de®ned by the coordinate axis of Figure 3.16, the sign change of the HTP and De can only appear if the perturbing atom crosses one of the mirror planes of the skeleton. Under this condition an unequivocal relation exists between the signs of the HTP and De. The chromophore, i.e., the skeleton, determines De and thus is the internal label. The phase determines the HTP and thus is the outer label. Here it is important to mention that the chromophore, interpreted geometrically, is the region within the molecule that is responsible for the light absorption. In Figure 3.17 the model is presented for an aminoanthraquinone where the contributions of all perturbing atoms have to be summed. A corresponding ``mean®eld'' description has been developed [68] and such an HTP/De relation has been experimentally proven [14], [67].

90

H.-G. Kuball and T. HoÈfer Figure 3.17. Intermolecular chirality transfer. Model for developing a sector rule for the HTP.

3.4.2

Phenomenological Descriptions

There are three types of approaches to describe the HTP quantitatively. From the phenomenological point of view a distortion of material in the surroundings of a chiral point leadsÐaccording to the elasticity theoryÐto an elastic twist and thus, to a cholesteric phase [69]. The second approach starts from molecular interactions by which a mean ®eld potential with a suitable symmetry can be constructed. In the resulting ``models for cholesteric phases'' the size and sign of the pitch depend on the potential parameters chosen [64]. Suitably chosen chiral potentials allow us to simulate chiral phases by the Monte Carlo technique. Even in a system of hard ellipsoids a phase with a suprastructural chirality (twisted nematic) can be obtained through induction by apparent external forces given by twisted boundary conditions [70]. The third approach starts from a qualitative discussion of the packing of the molecules. Heppke et al., e.g., in [2], assume for guest/host systems chiral conformers of the guest whereas Gottarelli et al. [54]±[57] assume chiral conformers of the guest and of the host. With these models chirality is introduced into the solvent molecules by forcing them to form chiral conformers. This concept resembles the Pfei¨er e¨ect for induced optical activity and CD in isotropic solutions [71]±[73]. To check this concept of chiral conformers of the host, an interesting experiment was proposed where a racemic achiral compound with a 1:1 ratio of chiral conformers, possessing a racemization barrier smaller than kT, was dissolved in an induced cholesteric phase. It was expected that an induced shift of the equilibrium of chiral host conformers to an enantiomeric excess should enhance the chiral induction. In spite of the fact that such phenomena should appear, these experiments have not proven this prediction successfully, as yet [74]±[76].

3. From a Chiral Molecule to a Chiral Anisotropic Phase

3.4.3

91

Chiral Liquid Crystal Phases by Monte Carlo Simulations

In order to describe the phase structures based on a direct calculation of the intermolecular interaction the Monte Carlo simulation can be used successfully. The total intermolecular interaction energy U…Wi ; Wj ; rij † between two chiral molecules i and j can be split into two parts, namely, the energy of achiral interaction, aUa …Wi ; Wj ; rij †, and the energy of chiral interaction, cUc …Wi ; Wj ; ^rij †: U…Wi ; Wj ; rij † ˆ aUa …Wi ; Wj ; rij † ‡ cUc …Wi ; Wj ; rij †:

…3:15†

The pseudoscalar energy contribution Uc …Wi ; Wj ; ^rij † changes sign if the interacting chiral molecules are replaced by their mirror images, the enantiomers. The totally symmetric chiral interaction energy contribution is then obtained by multiplying Uc …Wi ; Wj ; ^rij † with a pseudoscalar chirality parameter, denoted as c. If only principle e¨ects should be shown, the very handsome Gay±Berne potential is used extensively. If only uniaxial chiral molecules are discussed, in order to avoid the phenomenon of the intramolecular chirality transfer, it follows [77]: ui ; u^j ; rij † Ua …^ ( ˆ

"

4e…^ui ; ^uj ; ^rij †

s0 rij ÿ s…^ ui ; ^ uj ; ^rij † ‡ s0

12



s0 ÿ rij ÿ s…^ui ; ^uj ; ^rij † ‡ s0

6 #)

;

…3:16† where ^rij denotes the unit vector parallel to rij and rij ˆ jrij j. The orientation uj along the symmetry axes of the is described by the unit vectors ^ ui and ^ rotationally symmetric molecules. The explicit expressions for the orientation- and separation-dependent parameters s…^ui ; ^uj ; ^rij † and e…^ui ; ^uj ; ^rij † have to be chosen to determine the form of the Gay±Berne molecule [78]. The energy of chiral interaction, cUc …Wi ; Wj ; rij †, is given by ( )  7 s0 ui ; ^ uj ; ^rij † ‰…^ui  ^uj †
^rij Š…^ui
^uj † : Uc …^ui ; ^uj ; ^rij † ˆ ÿ 4e…^ rij ÿs…^ ui ; ^ uj ;^rij †‡s0 …3:17† This interaction term, which is proportional to the ®rst pseudoscalar term in a multipole expansion of the intermolecular interaction energy, as obtained by Goossens [79], [77], [80], was su½cient to obtain a rich polymorphism of chiral liquid crystal phases given by cholesteric and several blue phases in dependence on temperature and the chirality parameter c measuring the strength of the energy of chiral interaction, both for calamitic and discotic chiral Gay±Berne molecules [77], [81].

92

H.-G. Kuball and T. HoÈfer

3.4.4

A Concept for a Tensorial Description of the HTP

The experimental investigation of the chiral induction of the aminoanthraquinones and binaphthyls showed, on the one hand, that the induction by a dopant molecule depends strongly on the orientation of the orientation axis with respect to its skeleton and, on the other hand, on the orientation of a chiral group with respect to the principal axes of the order tensor of the molecule. It became clear that the quantitative contribution to the HTP of a chiral ligand of a molecule of class A depends on the orientation of this chiral ligand with respect to the skeleton. These properties are typical for the behavior of tensorial quantities and thus a tensorial description has to be developed. 3.4.4.1

The Surface Tensor and Helicity Tensor Description

Assuming that the chirality is determined by the surface of a molecule, Nordio et al. derived for the HTP [31], [32]: r 2 RTean X Qij Sij ; …3:18† HTP ˆ ÿ 3 2pk22 Vm i; j  ÿ dij † and Qij are the coordinates of a traceless order where Sij ˆ 12 …3gij33 tensor and a traceless helicity tensor, respectively. For the calculation of the order tensor of the guest it has been assumed that host molecules are bound with an anchoring energy ean to the surface of the guest molecule [37]. The twist elasticity constant k22 and the molar volume of the guest±host system Vm are taken from experimental measurements and corrected by a scaling factor for the anchoring energy ean . The HTP of various molecules has been calculated by (3.18) with helicity tensors resulting from an approximation of the surface of the guest by di¨erent techniques. Especially for atropisomers of the binaphthyl series the description has been very successful. A restriction of the theory is that the HTP is zero for isotropically distributed chiral guest molecules in an ordered achiral host phase in spite of the fact that chirality has to be maintained. This is a consequence of the fact that Qij is traceless and, therefore, the temperature-dependence of the HTP is not adequately described. Furthermore, de Gennes has shown within the approximation of the elasticity theory that a pseudoscalar parameter of a guest is su½cient to induce a cholesteric phase. In this case no anisotropy of the chiral induction can be obtained [69]. Because of that Nordio's and de Gennes' descriptions are two borderline cases.

3.4.4.2

The HTP and the Chirality Interaction Tensor Wij

If the chiral interaction of a ¯uid phase with a dopant molecule is described by an antisymmetric tensor of third rank, which can be reduced to a pseudotensor of rank 2, a description of the HTP can be obtained analogously to

3. From a Chiral Molecule to a Chiral Anisotropic Phase

93

that of the circular dichroism of anisotropic systems (ACD) [39], [82]: X gij33 Wij : …3:19† HTP ˆ i; j

The chirality interaction tensor Wij is responsible for the interaction of the chiral guest, the chirality of which P is described by the chirality tensor Cij , with the anisotropic host Wij ˆ k Cik Lkj . Lij covers the anisotropic host properties. W ˆ TrfWij g is di¨erent from zero. The gijkl are the orientational distribution coe½cients of the guest in the molecular ensemble of the guest± host phase. From the two possible representations to avoid non-diagonal elements in (3.19), the representation in the system of principal axes of the order tensor is chosen instead of the principal axes of the chirality interaction tensor Wij . With Saupe's order parameters the HTP is also a sum of three terms    W11 ‡ g2233 W22 ‡ g3333 W33 g HTP ˆ fg1133   1      1 1 p   …W22 ÿ W11 †D : ˆ 3W ‡ …W33 ÿ 3W †S ‡ 3

…3:20†

W=3 is the contribution of the chiral guest to the HTP in the case of an isotropic orientational distribution of the guest in the anisotropic guest±host phase. Contributions to the coordinates of Wii introduced by other traceless tensors are canceled in the trace of Wij . From (3.20) with W ˆ 0, the equationpof et al. [31] can be obtained, whereas with W33 ÿ 13 W ˆ 0 and  Nordio  …1= 3†…W22 ÿ W11 † ˆ 0 the approximation of de Gennes follows. In (3.19) and (3.20) the dependence on the variation of the host order is only indirectly taken into account as in the presentation of Nordio et al. The temperature dependence of the host orderÐespecially for the ``isotropic'' W=3Ðis partially neglected. Experimental experience shows that there is no drastic e¨ect. The chirality interaction tensor can be determined with (3.19) or (3.20)  …T† from the measured HTP…T† and the coordinates of the order tensor gij33 (Figure 3.7) by a multiple regression procedure. For convenience, the calculated HTP…T† for the binaphthyls R-B1 and R-B3 to R-B7, that ®ts the experimental HTP…T† within the limits of experimental error, are presented as a function of the order parameter S  in Figures 3.18, 3.20, and 3.21 which is equivalent to the presentation as a function of T  . For R-B4, R-B5, and R-B6 the trace W ˆ trfWij g is mainly determined by the coordinate W22 , whereas for R-B7 the coordinate W33 is decisive (Figure 3.18). The exceptional behavior of R-B7 is a consequence of the switching of the orientation axis which now lies along the C2 -axis, which is the bisector of the dihedral angle y, whereas it is almost parallel to the naphthyl±naphthyl bond for R-B4, R-B5, and R-B6 (Figure 3.19) [46]. Because of the switch of the orientation axis for R-B7, the order parameter D  is changed and therefore the curvatures of the HTP…T† for R-B4 and R-B7 are di¨erent (Figures 3.18 and

94

H.-G. Kuball and T. HoÈfer

Figure 3.18. The measured HTP (d) of the binaphthyls (a) R-B4 and (b) R-B7 in ZLI-1695, the calculated HTP (9) according to (3.20), the contribution of the S   term (v) and the D  term (e) as a function of S  …13 U g3333 U 1†. The measured    region is between the vertical lines. …W11 ; W22 ; W33 † is …ÿ1; ÿ202; 41† for R-B4 and …ÿ63; 19; ÿ246† for R-B7.

3.19). From the inspection of extrapolated curves in Figure 3.18 it can be concluded that a helix inversion is possible for R-B4 but not for R-B7 in spite of the equal chiral elements in the two cases. Furthermore, from (3.20) a helix inversion follows when the sign of W33 is di¨erent from the sign of W as shown for R-B4. According to (3.20) the temperature-dependence of the HTP is typically represented by the extrapolated curves in Figure 3.18 if the relation D ˆ f …S; d† (Figure 3.7) is approximately ful®lled. Whether the curvature is convex or concave depends on the value of D  whereas the mean slope is determined by the S  term of (3.20). According to this, the temperaturedependence of the HTP…S  †Ðif only one conformer existsÐis represented by the function D ˆ f …S; d† (Figure 3.7) pictured on a straight line with a slope is given by W33 ÿ 13 W . The curvature is modi®ed by p which  …1= 3†…W22 ÿ W11 † and thus, as often found, produces a relatively ¯at curve.

Figure 3.19. The di¨erent orientation axes to which the D  term belongs. From 1 H and 13 C NMR spectroscopy in an isotropic solution a group symmetry of C2 results for R-B1 to R-B7 because of the conformational equilibrium of R-B7.

3. From a Chiral Molecule to a Chiral Anisotropic Phase

95

Figure 3.20. The measured HTP of (a) R-B4 and (b) R-B7 in ZLI-1695 (d), the  W11 calculated HTP (9) according to (3.19), the contributions of the three terms g1133      1  (‡), g2233 W22 (), and g3333 W33 ( ) as a function of S …3 U g3333 U 1†. The measured region is between the vertical lines.

The presentation of the results with (3.19) instead of (3.20) leads to a quite di¨erent insight, namely, that a tensor coordinate Wii Ðindependent of its sizeÐproduces only an essential contribution to the measured HTP when it is connected with a large tensor coordinate of the order tensor. For R-B4  W22 term whereas (Figure 3.20(a)) the HTP is mainly determined by the g2233   for the HTP of R-B7 (Figure 3.20(b)) the g3333 W33 term is decisive. R-B5 and R-B6 behave analogously to R-B4, but R-B4 to R-B6 behave di¨erently  Wii for the analyzed from R-B7. In Figure 3.21 the largest products gii33 binaphthyls are shown. The nonbridged binaphthyl compounds R-B1 and R-B3 do not ®t in the frame of the given description because of the very small HTP values and the

P  Figure 3.21. The largest terms of the sum i gii33 Wii which contributes to the HTP  of R-B1 (9), R-B3 (v), R-B4 (d), R-B5 (e), R-B6 (x) …g2233 W22 †, and of R-B7 (‡)   …g3333 W33 †.

96

H.-G. Kuball and T. HoÈfer

sign change without a change of the absolute con®guration. A cisoid conformation follows from the negative exciton couplet of the 1 Bb transition in the CD spectra for R-B1 to R-B7 with the R-con®guration [46]. The amplitude of the couplet of the unbridged 1,1 0 -binaphthyls R-B1 and R-B3 is only about 75% of the amplitude of the couplet of the bridged compounds R-B4 to R-B7 and thus the dihedral angles for R-B1 and R-B3 are approximately equal to those of the bridged 1,1 0 -binaphthyls [46]. On super®cial consideration it would be expected that, therefore, the HTP should also be determined  Wii of the same size as found for the bridged biby one large product gii33 naphthyls. But the ¯at potential curve of the unbridged 1,1 0 -binaphthyls for a rotation around the naphthyl±naphthyl bond of about 90 G 30 in the gas phase, obtained by an AM1 calculation, allows a Large Amplitude Motion (LAM; [83], [84]). Thus, di¨erently solvated conformers with di¨erent dihedral angles about the equilibrium angle y0 can be expected. Then it is only an apparent contradiction that the amplitudes of the exciton couplets of R-1 to R-7 are similar and that the coordinates of the experimentally determined chirality interaction tensor Wii of the unbridged 1,1 0 -binaphthyls are not comparable to those of the bridged compounds. This should be a consequence of the di¨erent type of averaging for the CD and the chirality interaction tensor. The mean W ii values for R-B1 and R-B3 are obtained as a  …y†Wii …y† whereas the CD is only a mean over the weighted mean of gii33 spectroscopic property independent of the order of the molecules. The order parameters of the di¨erently solvated conformers of R-B1 to R-B3 in the cisoid and transoid forms are very di¨erent on account of their length to breadth ratios. Since there is no symmetry argument for Wii …y† to have a chiral zero between 0 U y U 180 , no sign change occurs in this interval and, therefore, the Wii …ya † should be of similar size as the Wii for the bridged  1,1 0 -binaphthyls. But the coordinate of the order tensor gii33 …ya † is expected to vary strongly because of the large changes in the length-to-breadth ratio on going from a cisoid to a transoid conformer. The experimentally found sign change for the HTP of R-B3 can also be explained within the scope of this interpretation [46]. Very high positive HTP values have been obtained for 1,1 0 -substituted unbridged binaphthyls with very large alkyl-biphenyl- or alkyl-cyclohexyl-phenyl substituents [5], [7] which e¨ectively introduce new chiral elements into the molecule. The orientation axis for these compounds lies parallel to the naphthyl±naphthyl bond direction [85].

3.5

Discussion and Conclusion

The HTP is a chirality measurement which results from the fact that the chirality of the guest molecule is mapped onto the phase. As long as a cholesteric phase is induced, the suprastructural chirality of the phase can be described as a P or M helicity. Important points are that the HTP depends

3. From a Chiral Molecule to a Chiral Anisotropic Phase

97

strongly on the orientation of the orientation axis of the guest molecule with respect to its skeleton and on the orientation of the chiral group with respect to the principal axes of the order tensor. Thus, the phase can ``see'' the chirality of the molecule from di¨erent directions as it is known from ACD spectroscopy. Furthermore, for molecules of class A, chiral groups contribute additively to the HTP. But one and the same group contributes only with one and the same value to the HTP when it possesses the same orientation with respect to the principal axis of the order tensor. These experimental results require a tensorial description. One approach of a tensorial description has been presented by Nordio et al. where the helicity tensor is traceless which means that there is no intermolecular chirality transfer if the chiral guest is isotropically distributed in the ordered host. This is in contrast to the description within the elasticity theory given by de Gennes where only a pseudoscalar quantity is responsible for the induction of a cholesteric phase and no anisotropy of chiral induction seem to be necessary for the description. The link between limiting cases of de Gennes and Nordio et al. can be obtained by (3.19) where the de Gennes parameter has to be interpreted as the trace of the chirality interaction tensor Wij . The HTP e¨ect is given as a sum of products of a quantity that determines the order and a quantity that is responsible for the chirality of the molecules. The often-appearing low and very similar temperature-dependence of the HTP curves is only determined by the coordinates of the traceless tensor Wii ÿ 13 W . Wii A 13 W for i ˆ 1; 2; 3 leads to a temperature-independent HTP. One of the mechanism of  Wii helix inversion can be explained by the possibility that the products gii33 possess di¨erent signs. Note that the chirality interaction tensor Wij with respect to its own principal axes determines the HTP because only Wij , the diagonal elements with respect to the principal axes of the order tensor, are important. As a consequence the rotation of the xi -axes in¯uences the HTP strongly. Here it has to be pointed out clearly that the whole discussion given above refers only to the contribution of one conformer because each conformer contributes with its own tensor Wij to the e¨ect. The description has been successfully applied for bridged binaphthyls where the dominating term of (3.19) is connected to the tensor coordinate that belongs to the principal axis of the order tensor lying parallel to the C2 rotation axis intersecting the dihedral angle of the rotation of the naphthyl groups about each other which introduces the chirality into the molecule. For ¯exible molecules not all questions are answered with this tensorial description. Therefore, further experimental studies as well as the analysis of the theoretical foundation of the presented tensor relation of (3.19) are necessary. It should also be checked whether pseudotensors of higher rank contribute to the HTP. The spontaneous formation of chiral phases from achiral compounds [28], [29] can be described within the scope of (3.19) only as long as chiral conformers are involved and need an extension, e.g., if only a suprastructural order of achiral species is decisive for the e¨ect.

98

H.-G. Kuball and T. HoÈfer

Acknowledgments. Financial support from the Deutsche Forschungsgemeinschaft, the Volkswagen Stiftung, and the Fonds der Chemischen Industrie is gratefully acknowledged.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

G. Heppke and F. Oestreicher, Z. Naturforsch. 32, 899 (1977). G. Heppke and F. Oestreicher, Mol. Cryst. Liq. Cryst. (Lett.) 41, 245 (1978). A. GoÈbel-Wunsch and G. Heppke, Z. Naturforsch. 34, 594 (1979). G. Heppke, F. Oestreicher, and G. Scherowsky, Z. Naturforsch. 37, 1176 (1982). G. Heppke, D. LoÈtzsch, and F. Oestreicher, Z. Naturforsch. 41a, 1214 (1986). G. Heppke, D. LoÈtzsch, and F. Oestreicher, Z. Naturforsch. 42a, 279 (1987). G. Heppke, H.-S. Kitzerow, D. LoÈtzsch, and Ch. Papenfuû, Liq. Cryst. 8, 407 (1990). Ch. Bahr, C. Escher, D. Fliegner, G. Heppke, and H. Molsen, Ber. Bunsenges. Phys. Chem. 95, 1233 (1991). H.-G. Kuball and G. Heppke, Liquid Crystals Today 5, 5 (1995). H.-G. Kuball, Liquid Crystals Today 9, 1 (1999). H.-G. Kuball, R. Memmer, and O. TuÈrk, in: Ferroelectrics (edited by S. Lagerwall and L. Komitov), 2000 in press. H.-G. Kuball, Th. MuÈller, H. BruÈning, and A. SchoÈnhofer, Mol. Cryst. Liq. Cryst. 261, 205 (1995). H.-G. Kuball, H. BruÈning, Th. MuÈller, O. TuÈrk, and A. SchoÈnhofer, J. Mat. Chem. 5, 2167 (1995). H.-G. Kuball and H. BruÈning, Chirality 9, 406 (1997). R.S. Cahn, C. Ingold, and V. Prelog, Angew. Chem. 78, 413 (1966). Houben±Weyl, Methods of Organic Chemistry, Vol. E21a, Stereoselective Synthesis (edited by G. Helmchen, R.W. Ho¨mann, J. Mulzer, and E. Schaumann), Thieme-Verlag, Stuttgart, 1995. F. VoÈgtle, Supramolekulare Chemie, B.G. Teubner, Stuttgart, 1992. B. Kiupel, Ch. Niederalt, M. Nieger, S. Grimme, and F. VoÈgtle, Angew. Chem. 110, 3206 (1998); Angew. Chem. Int. Ed. 37, 3031 (1998). Topics of Current Chemistry, No. 197, Dendrimers (edited by F. VoÈgtle), Springer-Verlag, New York, 1998. C. Yamamoto, Y. Okamoto, T. Schmidt, R. JaÈger, and F. VoÈgtle, J. Am. Chem. Soc. 119, 10547 (1997). G. Snatzke, Angew. Chem. 91, 380 (1979). G. Snatzke, Chemie in unserer Zeit 16, 160 (1982). G. Snatzke and F. Snatzke, in: Analytiker Taschenbuch, Band 1 (edited by H. Kienitz, R. Bock, W. Fresenius, W. Huber, and G. ToÈlg), Springer-Verlag, Berlin, 1980, p. 217. P.J. Collings, Mol. Cryst. Liq. Cryst. 292, 155 (1997). A.B. Harris, R.D. Kamien, and T.C. Lubensky, Phys. Rev. Lett. 78, 1476 (1997). T.C. Lubensky, A.B. Harris, R.D. Kamien, and G. Yan, Ferroelectrics 212, 1 (1998). Handbook of Liquid Crystals (edited by D. Demus, J. Goodby, G.W. Gray, H.-W. Spiess, and V. Vill), Wiley-VCH, Weinheim, 1998.

3. From a Chiral Molecule to a Chiral Anisotropic Phase

99

[28] T. Sekine, T. Niori, J. Watanabe, T. Furukawa, S.W. Choi, and H. Takezoe, J. Mater. Chem. 7, 1307 (1997). [29] G. Heppke and D. Moro, Science 279, 1872 (1998). [30] L.D. Barron, in: New Developments in Molecular Chirality (edited by P.G. Mezey), Kluwer Academic, Dordrecht, 1991, p. 1. [31] A. Ferrarini and P.L. Nordio, J. Chem. Soc. Perkin Trans. 2, 455 (1998). [32] A. Ferrarini, P.L. Nordio, P.V. Shibaev, and V.P. Shibaev, Liq. Cryst. 24, 219 (1998). [33] A. Ferrarini, G.J. Moro, and P.L. Nordio, Mol. Phys. 87, 485 (1996). [34] A. Ferrarini, G.J. Moro, and P.L. Nordio, Phys. Rev. E 53, 681 (1996). [35] A. Ferrarini, G.J. Moro, and P.L. Nordio, Liq. Cryst. 19, 397 (1995). [36] L. Feltre, A. Ferrarini, F. Pacchiele, and P.L. Nordio, Mol. Cryst. Liq. Cryst. 290, 109 (1996). [37] A. Ferrarini, G.J. Moro, L. Nordio, and G.R. Luckhurst, Mol. Phys. 77, 1 (1992). [38] Circular Dichroism: Principles and Applications (edited by K. Nakanishi, N. Berova, and R.W. Woody), VCH, New York, 1994. [39] H.-G. Kuball and A. SchoÈnhofer, in: Circular Dichroism: Principles and Applications (edited by K. Nakanishi, N. Berova, and R.W. Woody), VCH, New York, 1994, p. 85. [40] J.-X. Guo and D.G. Gray, Liq. Cryst. 18, 571 (1995). [41] K.J. Mainusch and H. Stegemeyer, Z. Phys. Chem. NF 77, 210 (1972). [42] L. Velluz, M. Legrand, and M. Grosjean, Optical Circular Dichroism, VCH, Weinheim, Academic Press, New York, 1965. [43] H.-G. Kuball, A. Strauss, M. Kappus, E. Fechter-Rink, A. SchoÈnhofer, and G. Scherowsky, Ber. Bunsenges. Phys. Chem. 91, 1266 (1987). [44] A. SchoÈnhofer, M. Junge, R. Memmer, and H.-G. Kuball, Chem. Phys. 146, 211 (1990). [45] R. Memmer, H.-G. Kuball, and A. SchoÈnhofer, Mol. Phys. 89, 1633 (1996). [46] H.-G. Kuball, O. TuÈrk, I. Kiesewalter, and E. Dorr, 1999 to be published. [47] G.R. Luckhurst, C. Zannoni, P.L. Nordio, and U. Segre, Mol. Phys. 30, 1345 (1975). [48] N. Harada and K. Nakanishi, Circular Dichroic SpectroscopyÐExciton Coupling in Organic Stereochemistry, University Science Books, Millvalley, 1983. [49] A. Moscowitz, K. Wellman, and C. Djerassi, J. Am. Chem. Soc. 85, 3515 (1963). [50] H.-G. Kuball, Th. MuÈller, and H.-G. Weyland, Mol. Cryst. Liq. Cryst. 215, 271 (1992). [51] Th. MuÈller, Dissertation, University of Kaiserslautern, 1994. [52] H.-G. Kuball, R. Kolling, H. BruÈning, and B. Weiû, SPIEÐThe International Society for Optical Engineering (Polish Chapter) 3318, 53 (1998). [53] G. Gottarelli and G.P. Spada, Mol. Cryst. Liq. Cryst. 123, 377 (1985). [54] G. Gottarelli, G. Proni, G.P. Spada, D. Fabri, S. Gladiali, and C. Rosini, J. Org. Chem. 61, 2013 (1996). [55] G. Gottarelli, G.P. Spada, R. Bartsch, G. SolladieÂ, and R. Zimmermann, J. Org. Chem. 51, 589 (1986). [56] G. Gottarelli, G.P. Spada, and G. SolladieÂ, Nouv. J. Chim. 10, 691 (1986). [57] G. Gottarelli, M. Hibert, B. Samori, G. SolladieÂ, G.P. Spada, and R. Zimmermann, J. Am. Chem. Soc. 105, 7318 (1983).

100

H.-G. Kuball and T. HoÈfer

[58] H.-G. Kuball, B. Weiû, A.K. Beck, and D. Seebach, Helv. Chim. Acta 80, 2507 (1997). [59] B. Weiû, Dissertation, University of Kaiserslautern, 1999. [60] I. Dierking, F. Gieûelmann, P. Zugenmaier, K. Mohr, H. Zaschke, and W. Kuczynski, Z. Naturforsch. 49a, 1081 (1994). [61] I. Dierking, F. Gieûelmann, P. Zugenmaier, K. Mohr, H. Zaschke, and W. Kuczynski, Liq. Cryst. 18, 443 (1995). [62] Y.R. Lin-Liu, Y.M. Shih, C.-W. Woo, and H.T. Tan, Phys. Rev. A 14, 445 (1976). [63] Y.R. Lin-Liu, Y.M. Shih, and C.-W. Woo, Phys. Rev. A 15, 2550 (1977). [64] A.N. Zakhlevnykh and M.I. Shliomis, Sov. Phys. JETP. 59, 764 (1984). [65] H.-G. Kuball, S. Neubrech, and A. SchoÈnhofer, Chem. Phys. 163, 115 (1992). [66] J.A. Schellman, J. Chem. Phys. 44, 55 (1966). [67] H.-G. Kuball and O. TuÈrk, Polish J. Chem. 73, 209 (1999). [68] M. Osipov and H.-G. Kuball (1999), submitted. [69] P.G. de Gennes and J. Prost, in: The Physics of Liquid Crystals, 2nd ed., Clarendon Press, Oxford, 1995, p. 284. [70] M.P. Allen and A.J. Masters, Mol. Phys. 79, 277 (1993). [71] P. Pfei¨er and K. Quehl, Ber. Dtsch. Chem. Ges. 65, 560 (1932). [72] L.D. Hayward, Chem. Phys. Lett. 33, 53 (1975). [73] P. Moser, Helv. Chim. Acta 51, 1831 (1968). [74] W. Shang and M.M. Labes, Mol. Cryst. Liq. Cryst. 239, 55 (1994). [75] M.M. Labes, V. Ramesh, A. Nunez, W. Shang, and X. Luo, Mol. Cryst. Liq. Cryst. 240, 25 (1994). [76] M.M. Green, D. Weng, W. Shang, and M.M. Labes, Angew. Chem. Int. Ed. Engl. 31, 88 (1992). [77] R. Memmer, H.-G. Kuball, and A. SchoÈnhofer, Liq. Cryst. 15, 345 (1993). [78] J.G. Gay and B.J. Berne, J. Chem. Phys. 74, 3316 (1981). [79] W.J.A. Goossens, Mol. Cryst. Liq. Cryst. 12, 237 (1971). [80] R. Memmer, H.-G. Kuball, and A. SchoÈnhofer, Ber. Bunsenges. Phys. Chem. 97, 1193 (1993). [81] R. Memmer, Ber. Bunsenges. Phys. Chem. 102, 1002 (1998). [82] A. SchoÈnhofer, H.-G. Kuball, and C. Puebla, Chem. Phys. 76, 453 (1983). [83] S. Canonica and U.P. Wild, J. Phys. Chem. 95, 6535 (1991). [84] K. Elich, R. Wortmann, F. Petzke, and W. Liptay, Ber. Bunsenges. Phys. Chem. 98, 126 (1994). [85] H.-G. Kuball, H. Friesenhan, and A. SchoÈnhofer, in: Polarized Spectroscopy of Ordered Systems (edited by B. Samori and E.W. Thulstrup), Kluwer Academic, Dordrecht, 1988, pp. 242, 391.

4

Chemical Structures and Polymorphism Volkmar Vill

Due to the important role of chirality in liquid crystals, a large number and variety of chiral chemical compounds have been developed. This chapter describes the most important molecular fragments and classes of chemical structures (Section 4.2) which provide both chirality and mesogenic properties. The form of chiral phases depends on the principles of the mesophase formation (Section 4.3). Some relations between the molecular chirality and the appearance of mesophase chirality are discussed and chiral dopants are classi®ed (Section 4.4). With respect to the mesophase behavior and to optical and electro-optical applications, it is important to know how the mesogenic chirality can be modi®ed, e.g., chemically by photoisomerization, or by changes of temperature or composition for certain suitable compounds (Section 4.5). Finally, chiral liquid crystals provide not only optical and electro-optical applications but also applications in Chemistry, e.g., as chiral solvents for synthesis, chiral stationary phases in chromatography, or chemical sensors (Section 4.6).

4.1

Introduction

Chirality is a fascinating subject for liquid crystal research [1]. Molecular chirality is transferred to macroscopic chirality and the resulting e¨ects are: (a) induction of polar properties without changing the phase type; (b) induction of new phases with helical ordering, which are continuously miscible with non-chiral phases; and (c) induction of new phases, which can only exist with chirality. A chiral dopant in a nematic phase can cause a helical pitch and the resulting phase is called cholesteric (Ch ˆ N*). This phase is continuously miscible with the nematic phase (E¨ect (b)). In some special cases, the helical pitch might be in®nitely long as found for the compensated cholesteric phase. Even this phase is di¨erent from the achiral nematic phase, because it has polar properties, i.e., pressure can induce chiral ordering (E¨ect (a)). A chiral 101

102

V. Vill

dopant can also induce a blue phase from a nematic phase. This phase is not miscible with the original nematic phase and this phase can only be found for chiral molecules (E¨ect (c)). Similar e¨ects can be described for other phases. The smectic A phase of chiral compounds may show electroclinic e¨ect (E¨ect (a)) or can be transferred to the TGBA phase (E¨ect (c)). The smectic C phase of chiral compounds may have helical ordering (E¨ect (b)), but can have polar ferroelectric properties (E¨ect (a)) even without helical structures. Further, a TGBC phase or a SCa phase might be induced (E¨ect (c)). All the macroscopic e¨ects of chirality need molecular chirality, and the chemical requirements and strategies will be discussed in the following sections.

4.2

Chemical Sources of Mesophase Chirality

Even the ®rst thermotropic liquid crystals (cholesterol benzoate and cholesterol acetate) were chiral molecules, and their chiral mesophases were observed [2]. Here, the chirality is taken from the chiral pool. Because liquid crystals are needed in gram scales for physical research and technical applications, they should be prepared by short synthetic pathways using easily available starting materials. Thus, synthetic strategies based on the chiral pool are more often used than asymmetric synthesis or chiral separation techniques. Good chiral sources might be judged by these categories: availability; mesogenity; diversity; polarity; stability; e¨ective mesogenic chirality; and other material properties.

4.2.1

Steroids

Steroids are very easily available in large amounts (cholesterol, cholestanol, sitosterol, stigmasterol). Often they are mesogenic themselves (cholesterol ester), thus they not only induce the chirality but also cause the mesogenic properties. Steroids are available in a very high diversity. A selection between di¨erent diastereomers and constitutional isomers can be used to optimize some properties or to make detailed studies on structure±property relationships. Steroids are unpolar compounds and therefore well miscible with other liquid crystals, but their chemical stability is low. The ``e¨ective mesogenic chirality'' is huge. Cholesteric phases, blue phases, and ferroelectric phases might be obtained. Long pitches, short pitches, and helical

4. Chemical Structures and Polymorphism

103

Figure 4.1. Cholesterol benzoate (``CB''), Phases: Cr 150.5 N* 182.6 is.

inversions can be found using these compounds. Other material properties like viscosity, optical anisotropy, etc., may limit the use of these materials. More than 100 di¨erent steroids have been tested for liquid crystals research [3], about 30 of them were tested in systematic studies [4], but only cholesterol has been used frequently. Cholesterol benzoate (``CB'', Figure 4.1) [5] was the ®rst chiral liquid crystal. Cholesterol chloride (``CC''), cholesterol nonanoate (``CN''), and cholesteryl oleyl carbonate (``COC'') have been reported in hundreds of papers.

4.2.2

Terpenoids

Terpenoids are as unpolar as the steroids, but normally they have bad mesogenic properties. Some unmodi®ed terpenoids like carvon have been used to induce long helical pitches in nematic phases. Esters of menthyl, isopinocampheyl, or other terpenoidal alcohols have been used for cholesteric and ferroelectric phases.

4.2.3

Amino Acids

Amino acids are highly polar. Thus, unmodi®ed amino acids can be dissolved in water but not in classical, thermotropic liquid crystals. Nevertheless, very di¨erent uses have been found for them. The chemical replacement of the NH2 group by chlorine or ¯uorine yields very interesting chiral building blocks for calamitic mesogens. ``C7'' (Figure 4.2) [6], [7] is the most commonly used compound obtained from these structures. The chiral unit ``CHClÐCHMe'' has interesting properties. A lateral substituent like Cl or Me normally disturbs mesogenic ordering and reduces phase transition temperatures. The direct neighboring of the units reduces the free rotation around the CÐC bond. This more ®xed conformation causes strong dipole moments, high spontaneous polarizations, and other valuable properties. On the other hand, sterical hindrance is reduced and even simple two-ring systems can yield liquid crystals.

Figure 4.2. C7, Phases: Cr 55 SC 55 SA 62 is.

104

V. Vill Figure 4.3. PBLG.

Completely di¨erent structures and properties are observed for polybenzylL-glutamate (``PBLG''), Figure 4.3 which is one of the most commonly used lyotropic cholesteric systems. The a-helix of the peptide backbone causes rodlike molecular shapes, which give chiral nematic orders in organic solvents.

4.2.4

Lipids

Most naturally occurring lipids are not chiral (para½ns, oleic acid, palmitin, etc.) or are mixtures of compounds which are di½cult to separate (lecithin, asymmetric glycerols). A few chiral fatty acids have been used, like methylated (Figure 4.4) [8] and alicylic (Figure 4.5) [9] acids. The chiral centers of fatty acids are often comformational ¯exible and they are not connected with large dipole moments. Also, the chemical modi®cation of the chiral centers is more di½cult than for amino acids and sugars. Thus, chiral lipids have only limited use in liquid crystal research.

Figure 4.4. Phases: Cr 56.3 SIA 59.5 SCA 108.5 SC* 109.4 A 119.2 is.

Figure 4.5. Phases: Cr 62.5 (SC 56.5 SA 58.9 N* 61.7) is, prepared from hydnocarpic acid.

4.2.5

Sugars

Sugars are a commonly used source for amphiphilic liquid crystals [10]. These materials show lamellar, columnar, and cubic phases, but chiral phases are very rarely observed. Thermotropic cholesteric phases are never observed and lyotropic cholesteric phases based on asymmetric micelles only in a few cases [11]. The bicontinuous cubic phase of these glycolipids may have macroscopic chiral ordering, but this has not been resolved hitherto [12], [13]. Thus, alkylated sugars are chiral compounds, but not e¨ective

4. Chemical Structures and Polymorphism

105

Figure 4.6. Hexadecyl-b-D-glucopyranoside, Phases: Cr 81.5 SA 147.5 is.

Figure 4.7. Tomatin.

chiral liquid crystals. Figure 4.6 shows hexadecyl-b-D-glucopyranoside, of which the double melting was already discoverd by E. Fischer in 1911 [14]. Much more complex compounds exist in nature, e.g., tomatin (Figure 4.7). This compound has only a smectic A phase in the pure state [10], but it can be used as highly e½cent chiral dopant for lyotropic nematic phases [15]. Chiral mesophases can be obtained from sugars by several strategies. Many cellulose derivatives show thermotropic and lyotropic cholesteric phases [16]. Peracylated sugars can be used as chiral dopants for discoid± nematic phases [17]. Also classical cholesteric and ferroelectric phases can be obtained from carbohydrate-based compounds [18]. In this case, chiral oxaheterocycles are prepared from sugars. Figure 4.8 shows a chiral twin compound prepared from mannitol [19].

4.2.6

Chiral Alcohols

S-2-methylbutanol is the most commonly used chiral building block of liquid crystals. A typical example is CB15 [20], shown in Figure 4.9. A statistical analysis of the phase transition temperatures between the wing groups Ð OÐCH2 ÐCHMeÐC2 H5 and ÐOÐC4 H9 is shown in Table 4.1. The methylation of the butoxy chain induces chirality but decreases the nematic± isotropic transition by about 43 K. Also, the formation of smectic phases is disturbed.

106

V. Vill

Figure 4.8. Chiral twin from mannitol, Phases: Cr 202.0 (S 155.0) is.

Figure 4.9. CB15, Phase: Cr 4.0 SA ÿ54 N* ÿ30 is.

Table 4.1. Comparison of the transition temperatures between liquid crystals bearing chiral chains and their analogous compounds bearing unsubstituted chains of the same length, calculated with LiqCryst [3]. Format: P : DT G sT …n ˆ aa† : P ˆ phase, DT ˆ temperature di¨erence, sT ˆ scattering of data, n: number of analyzed pairs. No

comparison

1

ÐOÐCH2 ÐCHMeÐC2 H5 /ÐOÐC4 H9

2

ÐOÐCHMeÐC6 H13 /ÐOÐC7 H13

3

ÐCOOÐCHMeÐC6 H13 /ÐCOOÐC7 H13

4

ÐOÐC3 H6 ÐCHMeÐC2 H5 /ÐOÐC6 H13

Temperature changes N : ÿ43:0 G 8:8 …n ˆ 46† SA : ÿ25:7 G 15:0 …n ˆ 21† SC : ÿ10:1 G 10:5 …n ˆ 21† N : ÿ91:3 G 20:2 …n ˆ 10† SA : ÿ81:8 G 30 …n ˆ 3† SC : ÿ70:6 G 18:7 …n ˆ 6† N: no data SA : ÿ48:4 G 3:3 …n ˆ 6† SC : ÿ16:5 G 12 …n ˆ 3† N : ÿ24:3 G 8:6 …n ˆ 29† SA : ÿ17:1 G 9:1 …n ˆ 20† SC : ÿ4:1 G 7:7 …n ˆ 27†

4. Chemical Structures and Polymorphism

107

Figure 4.10. MHPOBC, Phases: Cr 84 SCA 118.4 SCg 119.2 SC 120.9 SCa 122.0 SA 148 is.

Many chiral secondary alcohols have been prepared by reduction of ketons or by enzymatic discrimination of racemates. The enantiomeric purity of the compounds is often less than 100%, but in many cases not documented in the literature. The most commonly used secondary alcohol is 2-octanol. This compound can be obtained enantiomerical pure in both enantiomers. A typical compound prepared from 2-octanol is MHPOBC (Figure 4.10) [21]. A statistical analysis of the phase transition temperatures between the wing groups ÐOÐCHMeÐC6 H13 and ÐOÐC7 H15 is also shown in Table 4.1. The transition temperatures are strongly decreased: the nematic phase by 91 K and smectic C by 71 K.

4.2.7

Fluorinated Compounds

Fluorine substitutions at chiral C atoms can have some advantages, because ¯uorine causes only a small steric hindrance but induces large dipole moments. Chiral ¯uorine compounds cannot be obtained from the natural pool, thus asymmetric synthesis is required. Figure 4.11 shows two examples [22], [23].

Figure 4.11. Chiral ¯uorinated compounds.

4.2.8

Liquid Crystals with Chiral Rings

Many chiral alicyclic rings have been used for liquid crystals. Often, a chiral ring system is only documented by a few examples. Figure 4.12 shows some of them. The references can be found in [3].

108

V. Vill Figure 4.12. Liquid crystals with chiral rings.

4.2.9

Axial and Planar Chirality

In principle, all forms of chirality have been used for liquid crystal synthesis. This involves compounds with single chiral centers, multiple chiral centers, chiral axes, chirality planes, etc. For example, chiral methylidencyclohexanes can be found in [24], chiral allens in [25], and binaphthyls in [26].

4.3

Classi®cation of Chiral Mesophase Systems

Liquid crystals (LCs) can be divided into many di¨erent subclasses: thermotropic LCs, lyotropic LCs, polymeric LCs, etc. Chirality has a speci®c in¯uence in all these systems. Thus, a careful distinction is recommended. Chiral nonbonded interactions have only low energies. Thus, they are more dominating in systems with small forces between the molecules. Amphiphilic compounds have strong molecular interactions and chiral phases are less common. An exception is the lyotropic nematic phase. Here, micelles are acting like ``molecules'' to build up the mesophase. The interaction between the micelles is small and cholesteric phases can be formed easily.

4.3.1

Monophilic Calamitic Liquid Crystals

Monomeric rod-like compounds with nematic and smectic phases are the source for the most commonly used and analyzed chiral phases, e.g., BP, Ch, TGBA , SC , SCa , etc. Most of the theories and statements of this chapter are specialized for this class of compounds.

4.3.2

Natural Polymers

Cellulose, chitin, DNA, silk, and many other natural polymers, including their simple derivatives, can show chiral mesophase behavior.

4. Chemical Structures and Polymorphism

4.3.3

109

Synthetic Polymers

Polymers can have similar chiral e¨ects as the corresponding monomers. Chiral centers in the wing groups or in the mesogenic groups can cause cholesteric and ferroelectric phases very similar to Section 4.3.1. However, polymers can have additional positions and e¨ects of chirality. Chirality can be located in the backbone or in spacer groups, chiral dopants can be ®xed by copolymerization, and cholesteric networks can be ®lled with guest compounds, etc.

4.3.4

Discotic Systems

Recently, ferroelectric properties have been found in chiral columnar systems [27] and also discoid±cholesteric and discoid±blue phases have been found [17]. H. Bock describes chiral ferroelectric systems in Chapter 10.

4.3.5

Thermotropic Amphiphilic Systems

Lecithins and glycolipids are naturally occurring chiral compounds, which can create a big variety of mesophases (lamellar, hexagonal, cubic, ripple, gel phases, etc.). Classical chiral phases like cholesteric or ferroelectric smectic phases are hitherto not reported and, if they were found, they would not be very typical for this class of compounds. Some of the bicontinuous cubic phases may have enantiomeric pure chiral space groups. Further, lyotropic cholesteric and blue phases might be formed (see below). Ordered tilted lamellar phases have been reported for lecithin and other amphiphilic compounds, but the special e¨ects of chirality, like ferroelectric properties or helical order, are unknown. Amphiphilic compounds can form chiral tubes and nano-structures. These systems are rather crystalline than liquid crystalline and will not be discussed here.

4.3.6

Lyotropic Amphiphilic Systems

Lyotropic systems can show chiral phases in the form of cholesteric phases. Asymmetric micelles are the base unit for these phases. Lyotropic blue phases have also been reported [28]. Detailed descriptions of chiral lyotropic phases are given by K. Hiltrop in Chapter 14.

4.3.7

Chromonic Systems

Chromonic liquid crystals are a special class of lyotropic liquid crystals. Flat aromatic systems with polar substituents form columnar aggregates by p±p interactions in solution. The mesophases of concentrated DNA solutions or of copper±tetracarboxyphthalocyanine are examples for this. Special types of chiral mesophases have been observed [29].

110

4.3.8

V. Vill

Biosystems

Natural occurring polymers can create cholesteric phases in vitro. The research of the relevance of chiral mesophases in vivo is a challenge for the future. Biological membranes are built by chiral molecules, but diastereomeric interaction with chiral guest molecules seems to be of little importance.

4.3.9

Free-Standing Films

Thin ®lms of chiral mesophases and particularly free-standing ®lms can behave quite di¨erently from bulk systems. Some properties of free-standing ®lms are discussed by P. Pieranski in Chapter 2.

4.4

Phenomenological Classi®cation of Chiral Dopants

The physical properties (e.g., pitch, spontaneous polarization) are independent of the chemical origin of the chirality (see Sections 4.2.1 to 4.2.10). Exact models concerning the e¨ects of molecular chirality are di½cult to ®nd and to quantify. Further, each chiral parameter may have independent rules. The helix of the cholesteric phase is oriented perpendicular to the main axis of the molecules, whereas the helix of the SC phase is oriented parallel to the molecular axis. The spontaneous polarization also depends on the strength of the dipole monents connected to the chiral centers. Direct quanti®cation of the di¨erent chiral e¨ects from the chemical structure is a big challenge for the future. An example for molecular models of chiral properties might be the Boulder model of the spontaneous polarization [30]. As starting point for a molecular model, some statements will be given here: (a) Mesophase chirality is often created by asymmetry of the molecular surface relative to the main axis of the molecule. (b) Chiral dopant systems are easier to describe than bulk chiral systems. (c) Strong helical twisting power and good mesogenic properties are opposing properties. (d) The order parameter of chiral molecules is correlated with the strength of mesogenic chiral e¨ects. Chiral dopants can be classi®ed according to Figure 4.13. Compounds of type C1, which have no mesogenic groups and only chiral centers, can induce chiral mesophases. Resulting pitches are long and spontaneous polarizations are small, because the chiral dopant has only a small order parameter and the transfer of its chirality to the host system is only small as well. Compounds of type C2 have at least one mesogenic group, which interacts with the director of the mesophase. The chirality is located outside the mesogenic group in a ¯exible spacer. C2a has only one mesogenic group, which

4. Chemical Structures and Polymorphism

111

Figure 4.13. Classi®cation of chiral dopants: (a) ``Dopant Type I'' (Stegemeyer [34]), (b) ``Dopant Type II'' (Stegemeyer [34]).

needs to be oriented parallel to the director. The calculation of chiral properties by ab initio methods might be di½cult, but empirical data can be used to give good approximations for helical twisting power and spontaneous polarization. The values are bigger, if the chiral centers are close to the core. Shifting of the chiral center outside follows an odd±even e¨ect. The good statistical basis of this dopant type has been used to analyze natural chiral alcohols [35]. Type C2a can cause strong chiral e¨ects and the molecules are often mesogenic themselves.

112

V. Vill Figure 4.14. Carvon, an example for dopant type C1.

Type C2b has two (or more) mesogenic groups that want to be parallel to the director, but they are twisted by a chiral spacer. These compounds can cause very strong chiral e¨ects, but the molecules are often not mesogenic themselves. If the mesogenic axes are almost parallel, then helical inversions can be induced by small conformational changes. A simple approximation of the molecular helical twisting power b of a chiral dopant in a cholesteric phase can be given for systems of type C2b with only two mesogenic groups by formula (4.1) [38]. a is the dihedral angle between the two mesogenic groups, d is the e¨ective distance between the layers of the mesogenic groups, S0 is the order parameter of the host, and S is the order parameter of the mesogenic groups of the guest bˆ

a S2
: 2pd S02

…4:1†

Molecular models are much more di½cult for type C3, where the chiral centers are located inside the mesogenic axis. Small changes of the orientation of the main axis of the molecule can cause large changes of the chiral properties, thus, they are very sensitive to small changes of pressure, temperature, or concentration. Inversion of the helix [33] or of the spontaneous polarization [34] is quite common. Even cholesterol chloride and iodide have a di¨erent sense of the cholesteric helical pitch [36], and the ®rst observation of helix inversion was done with a steroid compound [37].

4.5

Chemical Modi®cations of Mesogenic Chirality

Tunable and/or switchable chirality can be used for scienti®c and technical applications. The active parameter to switch chirality can be light, concentration, or temperature. Most commonly used is the photoisomerization of

Figure 4.15. 2,5-Di-O-(4-hexoxybenzoyl)-1,4:3,6-dianhydro sorbitol, an example for dopant type C2b.

4. Chemical Structures and Polymorphism

113

azo compounds [39]. Chiral centers are normally saturated aliphatic groups, whereas chromophors are usually unsaturated and aromatic. Thus, photoactive groups and chiral centers do not interact very much.

4.6

Applications of Mesophase Chirality in Chemistry

Chiral phases can serve for technical applications, e.g., re¯ecting colors or ferroelectic displays, but they can also be used for scienti®c applications in organic chemistry. Asymmetric synthesis and racemic separations are important tasks in chemistry. One would expect that cholesteric phases are good solvents for stereoselective reactions and good stationary phases for gas chromatography. The obtained results are, up to now, moderate. This can be explained by a simple model: liquid crystals are de®ned by a long-range orientational order, whereas reactions need a short-range order. A helical pitch of about 500 nm still causes only a twist of less than 1 between two molecules and this is not signifcant for chiral separations. Liquid crystals can be used as sensors for stereoelectronic e¨ects of molecules or as detectors of chirality. Even a small enantiomeric excess of less than 1% can create a texture change in a nematic phase and a measurable pitch. Dopants of type C2a can be used to determine the absolute con®guration of chiral centers and changes of aggregation or conformation of molecules can be displayed by pitch changes. More common is the use of lyotropic cholesteric phases as solvent for nmr studies [40]. Organic chemists prepare chiral compounds usually as a service for physicists and engineers. It is a challenge for organic chemists to use chiral phases themselves!

References [1] J.W. Goodby, J. Mater. Chem. 1, 307±318 (1991). [2] H. Stegemeyer, Nachr. Chem., Tech. Lab. 36, 360±364 (1988). [3] V. Vill, LiqCryst 3.2ÐDatabase of Liquid Crystals, LCI Publisher, Hamburg, 1998; Fujitsu FQS, Fukuoka, 1998; http://liqcryst.chemie.uni-hamburg.de. [4] T. Thiemann and V. Vill, J. Phys. Ref. Chem. Data 26, 291±333 (1997). [5] F. Reinitzer, Monatsh. Chem. 9, 421 (1888); O. Lehmann, Z. Phys. Chem. 4, 462 (1889). [6] R. Higuchi, M. Honma, T. Sakurai, and N. Mikami, EP 159.872; JP 84±74.748 (1985). [7] C. Bahr and G. Heppke, Mol. Cryst. Liq. Cryst. Lett. 4, 31 (1986). [8] G. Heppke, D. LoÈtzsch, M. Morr, and L. Ernst, J. Mater. Chem. 7, 1993±1999 (1997). [9] P. Delavier, K. Siemensmeyer, S. Rohde, V. Vill, and N. Weber (BASF), Ger. O¨en. DE-OS 4.316.444.7 (19.5.93/24.11.94), 9 pp.; Chem. Abstr. 122, 278273 (1994).

114

V. Vill

[10] H. Prade, R. Miethchen, and V. Vill, J. Prakt. Chem. 337, 427±440 (1995). [11] Minden and Vill, to be published. [12] S. Fischer, H. Fischer, S. Diele, G. Pelzl, K. Jankowsky, R.R. Schmidt, and V. Vill, Liq. Cryst. 17, 855±861 (1994). [13] P. Sakya, J.M. Seddon, and V. Vill, Liq. Cryst. 23, 409±424 (1997). [14] E. Fischer and B. Helferich, Liebigs Ann. Chem. 383, 68±91 (1911). [15] M. Pape and K. Hiltrop, Mol. Cryst. Liq. Cryst. A307, 155±173 (1997). [16] D.G. Gray, Carbohydr. Polym. 25, 277±284 (1994). [17] D. KruÈerke, H.-S. Kitzerow, G. Heppke, and V. Vill, Ber. Bunsen-Ges. Phys. Chem. 97, 1371±1375 (1993). [18] V. Vill, H.-W. Tunger, and M. von Minden, J. Mater. Chem. 6, 739±745 (1996). [19] V. Vill, U. Schimmel, and K. Siemensmeyer (BASF), Ger. O¨en. DE-OS 4.625.441 (25.6.96/2.1.98); Chem. Abstr. 128, 116526; Inter. Patent PCT-WO 97/49.694 (31.12.1997). [20] G.W. Gray and D.G. McDonnell, Electron. Lett. 11, 556±557 (1975). [21] A.D.L. Chandani, T. Hagiwara, Y.-I. Suzuki, Y. Ouchi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys., Part 2, 27, L729±L732 (1988). [22] M.D. Wand, W.N. Thurmes, R.T. Vohra, K. More, and D.W. Walba, Ferroelectrics 121, 219±223 (1991). [23] S. Nakamura and H. Nohira, Mol. Cryst. Liq. Cryst. 185, 199±207 (1990). [24] G. Solladie and R.G. Zimmermann, J. Org. Chem. 50, 4062±4068 (1985). [25] J. Stichler-Bonaparte, H. Kruth, R. Lunkwitz, and C. Tschierske, Liebigs Ann. 1375±1379 (1996). [26] G. Gottarelli, G.P. Spada, R. Bartsch, G. SolladieÂ, and R. Zimmermann, J. Org. Chem. 51, 589±592 (1986). [27] G. Heppke, D. KruÈerke, M. MuÈller, and H. Bock, Ferroelectrics 179, 203±209 (1996). [28] K. Radley, Liq. Cryst. 18, 151±155 (1995). [29] J. Lydon, Chromonics, Handbook of Liquid Crystals, Chap. 18, Wiley-VCH, New York, 1998. [30] M.D. Wand, R. Vohra, D.M. Walba, N.A. Clark, and R. Shao, Mol. Cryst. Liq. Cryst. 202, 183±192 (1991). [31] V. Vill, F. Fischer, and J. Thiem, Z. Naturforsch. 43a, 1119±1125 (1988). [32] G. Heppke, D. LoÈtzsch, and F. Oestreicher, Z. Naturforsch. 42a, 279 (1987). [33] V. Vill, M. von Minden, and D. Bruce, J. Mater. Chem. 7, 893±899 (1997). [34] H. Stegemeyer, R. Meister, H.-J. Altenbach, and D. Szewczyk, Liq. Cryst. 14, 1007±1019 (1993). [35] T. Ikemoto, K. Mitsuhashi, and K. Mori, Tetrahedron: Asym. 4, 687±694 (1993). [36] L.B. Leder, J. Chem. Phys. 55, 2649±2657 (1971). [37] H. Stegemeyer, K. Siemensmeyer, W. Sucrow, and L. Appel, Z. Naturforsch. 44a, 1127±1130 (1989). [38] V. Vill, to be published. [39] A.S. Angeloni, D. Caretti, C. Carlini, E. Chiellini, G. Galli, A. Altomare, R. Solaro, and M. Laus, Liq. Cryst. 4, 513±527 (1989). [40] L. Canet, A. Meddour, J. Courtieu, J.L. Canet, and J.J. Salaun, J. Am. Chem. Soc. 116, 2155±2156 (1994).

5

Cholesteric Liquid Crystals: Defects and Topology O.D. Lavrentovich and M. Kleman

This chapter reviews the basic static properties of defects in cholesteric liquid crystals. The elastic features of the cholesteric phase with deformations at short-range and long-range (as compared to the cholesteric pitch) scales are discussed. Spatial con®nement, together with the relative smallness of the twist elastic constant, often leads to twisted and thus optically active structures even when the liquid crystal is composed of nonchiral molecules. The application of topological methods is illustrated using the models of twisted strips, closed DNA molecules, and defect linesÐdisclinations and dislocations. The homotopy classi®cation of defects in cholesterics is similar to that in biaxial nematics, and predicts phenomena such as the topological entanglement of disclinations and the formation of nonsingular soliton con®gurations. The spatial con®nement of ordered structures (represented, for example, by cholesteric droplets suspended in an isotropic matrix) imposes certain restrictions on the con®gurations of the order parameter and requires the appearance of topological defects in the ground state. The layered structure of cholesterics leads to the formation of large-scale defects such as focal conic domains and oily streaks.

5.1

Introduction

Chiral liquid crystals belong to a wide class of soft condensed phases. The director ®eld in the ground state of chiral phases is nonuniform because molecular interactions lack inversion symmetry. Among the broad variety of spatially distorted structures the simplest one is the cholesteric phase in which the director n is twisted into a helix. The spatial scale of background deformations, e.g., the pitch p of the helix, is normally much larger than the molecular size ( p V 0:1 mm) since the interactions that break the inversion symmetry are weak. The twisted ground state of chiral liquid crystals willingly accepts the additional deformations imposed by external ®elds, surface interactions, or by a tendency of molecules to form smectic layers, hexagonal order, or doubletwist arrangements. Very often such additional deformations result in topo115

116

O.D. Lavrentovich and M. Kleman

logical defects. The complexity of twisted structures with defects makes the cholesteric liquid crystals an important subject to test the modern concepts of relationship between the symmetry of molecular interactions and macromolecular organization. The connection between symmetry and defects has been for decades at the very heart of physics [1]±[3]; nowadays, it becomes the subject of studies in biology. In this chapter we discuss the basic features of deformed structures in liquid crystals with chiral order. The characteristic scale of these deformations has to be compared to the scale p of ground deformations. Properties of defects and deformations that occur at scales smaller and larger than p are quite di¨erent. We start this chapter with a brief introduction to the elastic theory of cholesteric phases with the object of clarifying the di¨erence in description of short- and long-range deformations (Section 5.2). Section 5.3 discusses ``weak'' twist deformations. Weak twist deformations are not necessarily caused by the chiral nature of the liquid crystal molecules. The recent discovery [4] of chiral domains in smectics composed of achiral molecules con®rms the general thesis that chirality in soft-matter systems does not always require chiral centers in the molecules, see the paper by G. Heppke and D. Moro [5]. Examples of chiral bulk deformations can be seen even in much simpler nematic samples, where the symmetry is broken either because of the explicit action of the boundary conditions or because of a more subtle mechanism that involves the smallness of the twist elastic constant K2 . Section 5.4 explains the elementary topological concepts employing a model of twisted strips; related to these strips are closed DNA molecules. The homotopy classi®cation of line defects, disclinations, and dislocations, and its predictions (such as the topological entanglement of lines) are presented in Section 5.5. Homotopy theory de®nes the necessary conditions for the formation of defects by deducing the classes of possible defects from the symmetry group of the order parameter. Su½cient conditions are often provided by the spatial boundedness of the ordered media. In Section 5.6 we describe how the spatial con®nement of an ordered system leads to the appearance of defects in its equilibrium state. Section 5.7 reviews the topological solitons (or ``textures'') which are topologically stable but nonsingular. Section 5.8 discusses defects such as focal conic domains and the oily streaks provoked by the tendency of cholesteric layers to keep an equidistance in large-scale deformations. Finally, Section 5.9 is an look forward to possible further studies in the ®eld of defects in chiral liquid crystals.

5.2

Elastic Theory and the Hierarchy of Scales

We deal with situations where the director ®eld deviates from the ideal helix. There are two complementary approaches to describe distortions in the cholesteric phase, depending on the ratio L= p, where L is the characteristic

5. Cholesteric Liquid Crystals: Defects and Topology

117

scale of the deformations or the size of the liquid crystal sample. We distinguish weakly twisted cholesterics …L= p f 1† and strongly twisted …L=p g 1† cholesterics.

5.2.1

Weakly Twisted Cholesterics

In the absence of external ®elds or bounding surfaces, the equilibrium director con®guration of the uniaxial cholesteric phase has the form n…r† ˆ u cos j…r† ‡ v sin j…r†:

…5:1†

Here u and v are two mutually perpendicular unit vectors (with constant orientation in space) and j…r† ˆ q0 w r ‡ const;

…5:2†

where q0 ˆ 2p=p and w ˆ u  v is a unit vector along the helix axis [1]±[3]. The twisted con®guration (5.1) minimizes the free elastic energy density f ˆ 12 K1 …div n† 2 ‡ 12 K2 …n
curl n ‡ q0 † 2 ‡ 12 K3 …n  curl n† 2 ;

…5:3†

with splay …K1 †, twist …K2 †, and bend …K3 † terms; q0 is positive for a righthanded cholesteric, and negative for a left-handed cholesteric provided the trihedron …u; v; w† forms a right-handed coordinate system. For example, …nx ; ny ; nz † ˆ …cos q0 z; sin q0 z; 0† yields f ˆ 0 in the Cartesian coordinate frame for both q0 > 0 and q0 < 0. Expression (5.3) contains only the ®rst derivatives of the director. Since f is quadratic in ni; j , f @ …nk; j † 2 , the second derivatives ni; jk might bring comparable contributions to f. Invariant terms involving second derivatives are usually written as the sum of the mixed splay-bend …K13 † and saddlesplay …K24 † terms: f13 ‡ f24 ˆ K13 div…n div n† ÿ K24 div…n div n ‡ n  curl n†:

…5:4†

Although it is not di½cult to see that the saddle-splay term can be reexpressed as a quadratic form of the ®rst derivatives, div…n div n ‡ n  curl n† ˆ ni; i nj; j ÿ ni; j nj; i , we will keep the form (5.4) for subsequent discussion. The divergence nature of the terms (5.4) allows us to transform the volume inte„ gral … f13 ‡ f24 † dV into a surface integral by virtue of the Gauss theorem … …… div g dV ˆ z n
g dA; …5:5† A

where g ˆ …K13 ÿ K24 †n div n ÿ K24 n  curl n and n is the unit vector of the outer normal to the surface A. However, K13 and K24 must not be neglected on the grounds of transformation (5.5). Whatever the way of integration of f, f13 , and f24 , the resulting elastic energy scales linearly with the size of the deformed system. The di¨erence between f and … f13 ‡ f24 † is more subtle and shows up when one looks for an equilibrium director con®guration by min-

118

O.D. Lavrentovich and M. Kleman

„ imizing the total free-energy functional … f ‡ f13 ‡ f24 † dV : the K13 and K24 terms do not alter the Euler±Lagrange variational derivative for the bulk, but they can in¯uence the equilibrium director through the boundary conditions at the surface A (which might also be the imaginary surface of the defect cores). Since the procedure of inclusion of the K13 term into the minimization problem is still debated, we will not consider this term here. The K24 term will be preserved since it brings an important insight into the nature of some chiral structures, such as double-twist con®gurations. When L= p f 1, the cholesteric does not di¨er much from the nematic phase. No wonder therefore that optical observations for weakly twisted cholesterics reveal ``thick'' (nonsingular) and ``thin'' (singular) line defects Ðdisclinations similar to that in the nematic phase. Moreover, in droplets of the so-called ``compensated'' cholesteric mixtures with extremely small L= p one can observe point defects [6] which, from the topological point of view, are allowed only in a nematic phase. The behavior of weakly twisted structures depends on the relative values of the elastic constants in (5.3) and (5.4). As we shall see in the next section, splay and bend distortions are often relaxed by twist. It is therefore important to know the elastic constants for di¨erent types of deformations; these constants are speci®ed by molecular structures and interactions. Small Molecules Liquid Crystals (SMLC's). In most cases, K2 is small as compared to K1 and K3 . For example, for 5CB [7]: K1 ˆ 0:64  10ÿ6 dyn;

K2 ˆ 0:3  10ÿ6 dyn;

K3 ˆ 1  10ÿ6 dyn:

The coe½cient K24 is very hard to determine; recent studies reviewed by Crawford and Zumer [8] indicate that in nematics K24 @ K1 ; K3 . Liquid Crystal Polymers (LCP) [9]. One does not ®nd in this case the simplicity of SMLCs but, on the other hand, one expects that the coe½cients would relate in an interesting way to molecular conformationsÐa ®eld of research which is still open to investigation. A general result is that K2 remains smaller than K1 and K3 , and is of the same order of magnitude as SMLCs. This result is rather intuitive, since the molecular length does not play a priori an important role in a pure twist deformation. On the other hand, K1 and K3 are strongly modi®ed. In rigid polymers, K1 and K3 increase with molecular weight and K3 increases faster than K1 . In semi¯exible polymers, two features appear when the molecular weight increases. First, the molecular length l becomes larger than the persistence length L; this has an e¨ect on K3 , whose variation with l reaches a maximum when l > L. Second, the density of chain-ends decreases when l increases; this has a direct e¨ect on K1 . In the limit when the chains become in®nitely long, any splay deformation at constant polymer density is forbidden, and K1 becomes increasingly large. The chain ends contribute to the total energy by the elastic deformation they carry and by their entropy that gives rise to a large contribution to K1 .

5. Cholesteric Liquid Crystals: Defects and Topology

5.2.2

119

Strongly Twisted Cholesterics

At L=p g 1, the elastic properties of the cholesteric are close to that of the lamellar phases. Here again, two di¨erent situations are possible. First, the cholesteric layers might be only slightly bent and preserve the topology of ¯at surfaces. These small deformations can be described by a single scalar variable, the component of the displacement u…x; y; z† of the layers along the normal of the nonperturbed layers, taken as the z-axis. The free-energy density in terms of the layer dilatation and small tilts is [1]±[3]:  f ˆ

1 2B

!2  qu 2 1 q2u q2u ‡2 K ‡ ; qz qx 2 q y 2

…5:6†

where one introduces renormalized constants B ˆ K2 q02 and K ˆ 38 K3 ; note that this renormalization does not take into account divergence terms. When the deviations of layers from the ¯at geometry are substantial, the deformations are more appropriately characterized by the principal curvatures s1 ˆ 1=R1 and s2 ˆ 1=R 2 of the cholesteric layers [10]. The elastic free energy density can be cast in the form f ˆ 12 K…s1 ‡ s2 † 2 ‡ 12 Bg 2 ;

…5:7†

where g ˆ jp ÿ p0 j=p0 is a relative dilatation of the layers. Scaling arguments show that the curvature elasticity fc ˆ 12 K…s1 ‡ s2 † 2 and the ``positional'' elasticity fp ˆ 12 Bg 2 should be treated on di¨erent footings when L=p g 1. Let L be a typical length of the deformation that shows up in all three spatial directions. The corresponding energies are Fc @ KL 2

and

Fp @ BL 3 ;

…5:8†

hence Fp =Fc @ …L= p† g 1. In other words, at L=p g 1, the theory treats the cholesteric medium as a system of equidistant (and thus parallel) layers with predominantly curvature distortions. Generally, the boundary conditions can be satis®ed only by the appearance of large-scale defects, such as focal conic domains and oily streaks. The coarse-grain model (5.7) does not take into account the saddle-splay term fss ˆ Ks1 s2 , where G ˆ s1 s2 is the Gaussian curvature of layers, and K is the saddle-splay constant (di¨erent from K24 in (5.4)). The dependence of K on the Frank constants has not been calculated so far. Partially, the omission is justi®ed by the fact that the K term does not change when the layers experience small ¯uctuations around the basic topologies (¯at layers, cylinders, tori, etc.). Transitions between these geometries, such as nucleation of a focal conic domain in a system of ¯at layers, should involve the K term. According to the Gauss±Bonnet theorem, the integral of the Gaussian curvature over a closed manifold is a constant de®ned by the Euler characteristic E of the manifold

120

O.D. Lavrentovich and M. Kleman

‡ s1 s2 dS ˆ 2pE;

…5:9†

E ˆ 0 for a torus and E ˆ 2 for a sphere. This topological feature makes the Gaussian curvature insensitive to small elastic deformations and sensitive to topological changes. In addition, it is precisely the nonzero value of the integral (5.9) that brings topological defects into the ground state of the con®ned liquid crystals, such as suspended droplets, as discussed in Section 5.6.

5.3 5.3.1

Weak Twist Deformations Con®nement-Induced Twists

As was established a long time ago by Mauguin [11], pure twist deformations can be produced by placing a nonchiral nematic liquid crystal between two parallel rubbed solid surfaces and then rotating one plate in its own plane relative to the other. Such a structure is optically active despite the fact that the nematic molecules are not chiral. The twist is maintained by the surface ``azimuthal'' anchoring. One would expect that when the director is allowed to rotate in the plane of one of the plates (an ``isotropic'' plate with no azimuthal anchoring), the twist and optical activity would disappear. Surprisingly, this was not what Meyerhofer et al. observed by placing nematic droplets on a rubbed plate and letting the upper surface of the liquid free [12]. The sessile droplets clearly demonstrated signi®cant optical activity, even when there was no external electric or magnetic ®eld. The phenomenon might be explained if one takes into account that the free surface of a sessile drop is usually curved (except in a rare case of complete wetting) and tilted with respect to the horizontal supporting plate. The wedge geometry forces the director to align normally to the thickness gradient (say, along axis y in Figure 5.1) in order to reduce the amount of splay and thus to reduce the elastic energy. The phenomenon can be called a ``geometrical anchoring'' [13]. However, when the bottom plate is rubbed along any direction di¨erent from y, the competition between the two easy axes might result in twist. To show this, let us calculate the energy per unit area of the wedge, neglecting director distortions in the plane of the cell [13]. We parametrize the director through the polar angle y and the azimuthal angle j as …nx ; ny ; nz † ˆ …sin y…z† cos j…z†; sin y…z† sin j…z†; cos y…z††. At the bottom plate, y…z ˆ 0† ˆ p=2 and j…z ˆ 0† ˆ 0. The director is tangential to the upper surface. If the two bounding surfaces were parallel, then in equilibrium y…z† ˆ p=2 and j…z† ˆ 0. Suppose now that the upper surface is tilted around the y-axis by an angle g. The polar angle y…z ˆ d† now depends on g and on the azimuthal parameter j0 which is the angle between n and a ®xed axis x 0 in the inclined upper plane: y…z ˆ d† ˆ arccos…sin g cos j0 †. Small

5. Cholesteric Liquid Crystals: Defects and Topology

121

Figure 5.1. A tangentially aligned nematic liquid crystal con®ned between two plates; the bottom plate is rubbed, the top plate is isotropic. The tilt of the upper plate tends to reorient the director normally to the plane of the ®gure (geometrical anchoring). In combination with surface anchoring at the bottom plate, this results in the twist deformation.

deviations from the uniform state, y…z† ! p=2 ‡ y1 …z† and j…z† ! 0 ‡ j1 …z† lead to the free energy density f ˆ 12 K1 y1;2 z ‡ 12 K2 j1;2 z . The bulk equilibrium equations, y1; zz ˆ 0 and j1; zz ˆ 0, together with the boundary conditions above, lead to the energy per unit area    K1 K2 tan j0 2 ‰arcsin…sin g cos j0 †Š 2 ‡ arctan : …5:10† Fˆ 2d 2d cos g According to (5.10), the equilibrium azimuthal angle at the upper surface can be nonzero (Figure 5.2). This implies twist and hence optical activity of the sessile droplet. The twist angle increases as the ratio K2 =K1 decreases so that the e¨ect might be strongly pronounced for nematic polymers such as poly-g-benzylglutamate (PBG), where the ratio K2 =K1 can be as small as 0.1 or even smaller [14]. Twist relaxation of splay and bend is a general phenomenon in materials with small K2 . Chiral structures can occur in defective nematic samples even when there is no azimuthal anchoring at all. Twisted brushes observed by Press and Arrott in textures of lens-shaped nematic droplets ¯oating on the water surface are one example [15]. Another well-known illustration of twist relaxation is the periodic pattern of stripes that occur in the geometry of splay Frederiks transition in polymer nematics with a small (less than 0.33) ratio K2 =K1 [16]. A ®eld applied normally to the planar nematic cell causes stripe structures composed mostly of twist rather than the uniform splay response observed in regular materials. An especially clear demonstration of twist relaxation is given by tangentially anchored spherical nematic droplets suspended in an isotropic matrix (glycerin), Figure 5.3. The director lines join two point defectsÐboojums at

122

O.D. Lavrentovich and M. Kleman

Figure 5.2. Elastic energy versus azimuthal angle at the top surface for the nematic wedge, see text. The twist angle increases with the increase in tilt g and the decrease in the ratio K2 =K1 .

the poles of the droplet. However, instead of a naive picture, with lines being meridians that lie in the planes of constant azimuth, one observes a twisted structure [17]. The director lines are tilted with respect to the meridional planes. This tilt decreases as one approaches the axis of the droplet. As in the previous example, the twist replaces energetically costly splay [18]. Each droplet is optically active despite the nonchiral nature of the molecules of both the nematic and matrix. Of course, there is an equal number of ``left''and ``right''-handed droplets in the dispersion. The droplets shown in Figure 5.3 present in fact a double twist rather than a simple unidirectional twist. Double twist is discussed below in relation to the saddle-splay coe½cient.

5.3.2

Double Twist

One may inquire about the meaning of the K24 term in a weakly twisted cholesteric; the solution is in the double-twist tendency of cholesterics [19]± [21]. Let n0 be some director, e.g., along the axis Z in Figure 5.4. In the local state of the smallest energy, the chiral molecules in the vicinity of n0 tend to rotate helically along all the directions perpendicular to n0 . This double twist is energetically preferable to the one-dimensional twist, at least for some chiral materials. In cylindrical coordinates, the elementary double-twist con®guration is nr ˆ 0;

ny ˆ ÿsin c…r†;

nz ˆ cos c…r†;

…5:11†

5. Cholesteric Liquid Crystals: Defects and Topology

123

Figure 5.3. Double-twisted nematic droplets suspended in an isotropic matrix. The central part of the droplet is bright when the polarizers are crossed and one of them is aligned along the droplets' axes (a); the central part can be made dark by changing the angle between the polarizer and analyzer. This behavior indicates the optical activity of the droplets caused by the director twist. The insert shows the director con®guration at the droplet's surface. Nematic n-butoxyphenyl ester of nonylhydrobenzoic acid dispersed in glycerin [17].

with c…0† ˆ 0. The free energy is  2 qc 1 sin 4 c K24 d ÿ sin c cos c ‡12 K3 2 ÿ …sin 2 c†: f ˆ 12 K2 q0 ÿ qr r r r dr

…5:12†

There is no K13 term, since div n 1 0. Integrating f we see that the K24 term contributes to the energy of a cylinder of matter of radius R by the quantity

124

O.D. Lavrentovich and M. Kleman

Figure 5.4. Blue phases are composed of regions with double twist; three such regions, with a singularity that relieves frustration between them, are shown in the ®gure. Two cylinders with double twist match at the contact point if the director tilt at their surfaces is p=4; however, the region where all three cylinders meet is singular. In current models, such singularities form a network of disclination lines. The circle marks the ``core'' of the disclination; the insert shows the director lines around the core.

F24 ˆ ÿK24

…R 0

2p

d …sin 2 c†
dr ˆ ÿ2pK24 sin 2 c…R†; dr

…5:13†

which is negative for any value of c…R† 0 pn, when K24 is positive. The nucleation of a double-twisted cholesteric geometry is favored in such a case, in particular when K1 is large compared to K3 . Examples of double-twist geometry are nonsingular disclinations of strength k ˆ 2 in cholesteric spherulites (often observed in biopolymers) that are discussed later. Another geometry with double twist is met in the chromosome of microscopic algae, Prorocentrum Micans (dino¯agellate chromosomes), which have been studied by optical and electron microscopy techniques [22]. As proposed in [19] (see also [23]), the structure contains two k ˆ 12 disclination lines which rotate helically about the chromosome axis. The double-twist geometry has a limited size, beyond which double twist decreases and frustrations in the system become too large. The layers have a negative Gaussian

5. Cholesteric Liquid Crystals: Defects and Topology

125

curvature. This geometry is favored over the spherulitic geometry, probably when K1 is smaller than K3 , because the k ˆ 12 lines cause splay. Also, while the nucleation of the k ˆ 2 geometry is easier with K24 > 0, it is the contrary to the chromosome. For a more general discussion, see review [24] on the problem of frustration. Finally, if K24 is positive and very large, the cylindrical geometry can become stable versus the cholesteric phase: this is the origin of the blue phases (BPs). In Figure 5.4, as the distance from the Z-axis increases, the cholesteric cylindrical shells become ¯atter and the double twist smoothly disappears. The director far-®eld distribution becomes closer to the onedimensional twist of the usual cholesteric phase; the energy gain is reduced. Thus the double twist cannot be extended over the whole three-dimensional space. A typical radius of the energy-gaining cylindrical region about the n0 -axis is the half-pitch p=2. (This is the reason why we discuss the double twist as a weakly twisted structure; the situation should not be confused with the fact that the blue phases usually occur for small-pitch materials). Now these cylinders of ®nite radius cannot tile space continuously. According to the current models of blue phases, this frustration is relieved by defect lines (of disclination type), either regularly distributed, or in disorder. Figure 5.4 illustrates how three cylinders of double twist generate a singularity in the region where they merge. A word of caution should be said about the interpretations of planar disclination lines as a source of saddle-splay. There is no K24 nor K13 contribution to the elastic energy of a straight planar disclination of the Frank type, …nx ; ny ; nz † ˆ …cos kj; sin kj; 0†, where k is integer or half-integer. Both terms vanish when the energy density is integrated over the azimuthal angle around the disclination core. A nonvanishing saddlesplay energy might come from the regions where the disclinations cross or from point defects, if such are present. The blue phases of types BPI and BPII are modeled as regular networks of disclination lines with periodicity of order p. Indeed, the three-dimensional periodic structure of these phases is revealed in their nonzero shear moduli, their ability to grow well-faceted monocrystals and Bragg re¯ection in the visible part of the spectrum (which is natural since p is of the order of a few tenths of a micron). The third identi®ed phase, BPIII, that normally occurs between the isotropic melt and BPII, is less understood. It might be a melted array of disclinations. Note that although most blue phases have been observed in thermotropic systems, double-twist geometries are relatively frequently met in textures of biological polymers, like DNA. DNA, polypeptides (such as PBG mentioned above), and polysaccharides (such as xanthan) and many other biological and nonbiological polymers have a de®nite handedness due to the chiral centers. Rod-like long molecules of these materials in water solutions often crystallize into a hexagonal columnar phase so that the cross-section normal to the rods reveals a triangular lattice. Since the polymers are chiral, close hexagonal packing competes with the tendency to twist [25], [26]. Macroscopic twist can proliferate by

126

O.D. Lavrentovich and M. Kleman Figure 5.5. Coexistence of twist and close hexagonal packing in a system of chiral rods that form a twist grain boundary phase with lattices of screw dislocations; unidirectional twist perpendicular to the plane of the ®gure; redrawn following [27].

introducing screw dislocations into the system [27], [28], in a way akin to the twist grain boundary phases of chiral smectics [29], [30]. Two types of defectstabilized phases that combine close packing and twist are possible. One is a polymer tilt grain boundary phase, a direct analog of the twist-grain boundary phase, and a usual cholesteric with a unidirectional twist, Figure 5.5. Another is a Moire grain boundary phase, similar to the blue phases with double twist. In the center of a cylindrical element, there is a polymer rod; the neighboring polymers twist around it, preserving the hexagonal close packing; the cylinders are packed together thanks to the honeycomb lattice of screw dislocations [27]. For a detailed discussion of the frustrated phases, such as blue, TGB, and chiral columnar phases, see the chapters by Bock, Crooker, Kitzerow, and Pieranski.

5.4

Twisted Strips

An adequate description of defects in ordered condensed media requires introducing a special mathematical apparatus, viz. the theory of homotopy, which is a part of algebraic topology. It is precisely in the language of topology that it is possible to associate the character of the ordering of a medium and the types of defects arising in it, to ®nd the laws of decay, merger, and crossing of defects, and to trace out their behavior during phase transitions, etc. The key point is occupied by the concept of a topological invariant, often also called a topological charge, which is inherent in every defect. The stability of the defect is guaranteed by the conservation of its topological invariant. The following simple example of twisted ribbon strips gives a ¯avor of the concept of a topological invariant.

5.4.1

Topological Charges Illustrated with Twisted Strips

Consider a set of closed elastic strips. Each strip is characterized by a number k that counts the number of times the ends of the strip are twisted by 2p before they are glued together to produce a ring, Figure 5.6. The ring with

5. Cholesteric Liquid Crystals: Defects and Topology

127

Figure 5.6. Topologically di¨erent rings of elastic strips: (a) nontwisted ring, k ˆ 0; (b) MoÈbius strip, k ˆ 12 ; (c) twisted strip with two di¨erent edges, k ˆ 1; and (d) twisted strip with two twists of opposite sign, k ˆ 0.

k ˆ 12, Figure 5.6(b), is the well-known MoÈbius strip. The deformation energy stored in any twisted strip is larger than the pure bend energy of the k ˆ 0 ring. However, to transform a twisted strip into a state k ˆ 0, one needs to cut the strip. There is no continuous deformation that transforms one strip into another if the two have di¨erent k's. The energy needed to cut the ribbon, Fcut @ US=a 2 , is much higher than the stored twist energy Ftwist @ k 2 KS=L; here L is the length of the strip, S is its cross-sectional area, and K @ U=a is some elastic constant of the order of the intermolecular energy; a is the molecular scale. Transitions between the states with di¨erent k's are prohibited by high-energy barriers. The allowed values of k are de®ned by the inner symmetry of the strip. For example, if the edges of the strip are di¨erent, e.g., marked by a thin line and a thick line, Figure 5.6(c), then only integer k's (2p-twists) are allowed. The quantity k does not change under any continuous transformation and is a useful invariant to label topologically di¨erent states. Left and right twists can be distinguished by the sign of k. Obviously, one can create a pair of left and right twists without cutting the strip, Figure 5.6(d); what matters is the total sum of k's, which should be preserved. Therefore, topological charges k's obey a conservation law. ``Topological twists'' considered above obey the following rules, that generalize to all types of topologically stable con®gurations: (1) defect types are related to the type of ordering of the system; (2) defects are characterized by quantized invariants (topological charges) such as k; and

128

O.D. Lavrentovich and M. Kleman Figure 5.7. Disclination line with a ptwist of the director ®eld.

(3) merger and decay of the defects are described as certain operations (e.g., additions) applied to their charges; conservation laws of topological charges control the results of merger and decay. The topological invariant k's form groups. The topological stability of twisted strips is similar to that of topological solitons; the issue of a singular core is not involved. However, one can draw a parallel between the twisted strips and singular defects, too. Imagine a circle around a p-disclination in a uniaxial nematic liquid crystal, Figure 5.7. The set of molecules centered in this circle form a MoÈbius strip with k ˆ 12. After going once around the circle, the director n ¯ips into ÿn, which is possible, since the nematic bulk is centrosymmetric, n 1 ÿn. The number k would remain equal to 12 whether the radius of the circle is taken larger or smaller, Figure 5.7. Thus the overall director con®guration can be characterized by k ˆ 12 (k is often called the ``strength'' of the disclination). At the disclination core, one faces the singularity: when the circle shrinks into a point, there is an in®nity of director orientations at this point. This rule of exact transformation n ! ÿn does not change if the nematic is replaced by a cholesteric.

5.4.2

DNA Loops

Twisted strips with di¨erent k's are of relevance to the problem of con®guration and replication of double-stranded DNA molecules. Two strands are arranged in a helicoid fashion in which a 2p-twist occurs per every 10.5 base pairs. In many organisms ranging from viruses and prokaryotes to some eukaryotes, DNA molecules form closed loops. Topologically, these loops remind us of a twisted strip with two distinctive edges, carrying a special integer Lk, referred to as the linking number of the two edges. It is an algebraic (i.e., accounting for the direction) number that shows how many times one (line) edge crosses a surface spanning the other (line) edge [31]. Lk

5. Cholesteric Liquid Crystals: Defects and Topology

129

is peserved in any conformational change of the DNA molecule that does not break the strands. If Lk is close to Lk0 ˆ l= p (l is the total DNA length, and p A 3:4 is the helix pitch), the DNA ring is elastically relaxed and can lie ¯at on a planar surface without contortion. Often Lk 0 Lk0 : the ends of the relaxed linear DNA duplex might be additionally twisted (or untwisted) by some number of rotations G2p before forming the ring. There are two ways of dealing with the induced strain. First, the number of base pairs per pitch can be changed; the ring remains planar and the linking number is equal to the number of turns of one strand around another. In that case, Lk ˆ k, the topological twist de®ned above. Second, the duplex axis can twist upon itself, leaving the number of pairs per pitch una¨ected. Such a supertwisted DNA is no longer planar and coils in three dimensions, like a buckled twisted ribbon. Whatever the case, while k and Lk stay una¨ected, and are still equal integral numbers of a topological nature, the global geometry (and consequently the energy of the ``twisted'' ribbon and the way it relaxes) depends on the elasticity properties of the molecule and is better described by two geometrical parameters: the (so-called) twist Tw and the writhe Wr. The twist can be written as ‡ 1 W…s† ds; …5:14† Tw ˆ 2p where W…s† is the rate of wrapping of either strand about the duplex axis (the angle of rotation of the base pairs the per unit length of the strand). This quantity can be de®ned equally for an open strip; Tw can take any value and we can refer to it as the geometrical twist. However, if the duplex axis is planar, one gets Lk ˆ Tw ˆ k. The writhe Wr of a curve C is a much more subtle quantity. Introduced by Fuller [32], it is the number of averaged selfcrossings (with sign) of the planar orthogonal projections of C (closed or not); in the DNA context it describes the buckling of the duplex axis, so to speak. Like Tw, Wr can take any value. We have the important relation Lk ˆ Tw ‡ Wr

…5:15†

with Lk (for two oriented curves C and C 0 ) and Wr (for an oriented curve C) given by double integrals: …… 1 r…s† ÿ r…s 0 † z :‰ds  ds 0 Š; Lk ˆ 4p C; C 0 jr…s† ÿ r…s 0 †j 3 …5:16† …… 1 r…s† ÿ r…s  †  z :‰ds  ds Š: Wr ˆ 4p C; C jr…s† ÿ r…s  †j 3 Here C is the duplex axis, say, and C 0 is any one of the strands. Wr vanishes when C is planar. Applications of these mathematical concepts to the elasticity of DNA can be found in [33]. To separate the DNA strands during replication, one needs to change the number Lk. It can be done directly by topoisomerases that cut one or both

130

O.D. Lavrentovich and M. Kleman

strands. In other cases, the replication occurs through local binding of the DNA molecule to proteins that creates zones of negative and positive supertwisting; for more details, see [34].

5.5

Line DefectsÐDisclinations and Dislocations

Generally, the order parameter of an ordered medium is a function of coordinates, c…r†. Distortions of c…r† can be of two types: those containing singularities and those without singularities. At singularities, c is not de®ned. For a three-dimensional (3D) medium, the singular regions might be zerodimensional (points), one-dimensional (lines), or two-dimensional (walls). These are the defects. Whenever a nonhomogeneous state cannot be eliminated by continuous variations of the order parameter (i.e., one cannot arrive at the homogeneous state), it is called topologically stable, or simply a topological defect. If the inhomogeneous state does not contain singularities, but nevertheless is not deformable continuously into a homogeneous state, one says that the system contains a topological con®guration (or soliton). The twisted wedge geometry considered in Section 5.2.1 is topologically trivial and equivalent to a uniform nematic. In contrast, point defects in droplets or defects involved in the formation of blue phases and twist grain boundary phases are topologically stable. The topological classi®cation of line defects in ordered media is based on the concepts of the order parameter (OP) space R and homotopy groups of the OP space [35]±[38]. Line defects are described by the so-called ®rst (or fundamental) homotopy group p1 …R†. Topological invariants labeling different defect lines are elements (or classes of elements) of p1 …R†. Point defects are described by the second homotopy group p2 …R†. This group is trivial for cholesterics, so that there is no topological point defect. Below we consider only line defects. We start with a uniaxial nematic, for which the predictions of topological classi®cation are rather simple (the results can be applied to a weakly deformed cholesteric, such as a 90 twisted nematic cell).

5.5.1

Disclinations in the Uniaxial Nematic

The OP space R is the manifold of all possible values of the OP that do not alter the thermodynamical potentials of the system. The energy of condensation Fcond takes a minimum value on R. For a uniaxial nematic, the OP space is a sphere of unit radius: any point on the sphere corresponds to a di¨erent orientation of the director n. Furthermore, since n 1 ÿn, any two diametrically opposite points on the sphere describe not just energetically equivalent states, but rather indistinguishable states. The unit sphere with identi®ed antipodal points is denoted S 2 =Z2 ; it is the OP space of a uniaxial nematic. Suppose that the nematic is deformed so that the director becomes a

5. Cholesteric Liquid Crystals: Defects and Topology

131

Figure 5.8. Continuous transformation of a disclination k ˆ 12 into a disclination k ˆ ÿ 12 in real space (director con®gurations above) and the corresponding transformation of contours in the OP space S 2 =Z2 .

function of the coordinates. The function n…r† maps the points r occupied by a sample in real space into the OP space. In the classi®cation of line defects, the mappings of interest are those of oriented contours encircling defects in real space. For example, in Figure 5.8, the contour g1=2 is mapped into the contour G1=2 in the OP space. The mapping n…r† of a closed contour in real space produces in the OP space S 2 =Z2 a contour which belongs necessarily to one of the following classes: either (a) a closed loop, or (b) a contour that connects two diametrically opposite points; this contour is closed because the two end points are identical. The contours of type (a) can shrink into a point; they correspond to disclinations of integer strength k which are topologically unstable and can be smoothly transformed into a uniform state through the well-known process [39], [40] that R.B. Meyer called ``escape in the third dimension'' [40]. The contours of type (b) are not contractible to a point under any continuous deformations, since the ends of the contours have to remain ®xed at diametrically opposite points. These contours correspond to disclinations of half-integer strength k. It is easy to see that all the contours corresponding to half-integer k's can be smoothly transformed one into another, Figure 5.8. The classes (a) and (b) of contours form a group Z2 of two elements, say, 0 and 12. The group operation laws are simply 12 ‡ 0 ˆ 12 and 12 ‡ 12 ˆ 0. The group of closed con-

132

O.D. Lavrentovich and M. Kleman

tours is called a fundamental group of the OP space and denoted p1 …R†. For a uniaxial nematic p1 …S 2 =Z2 † ˆ Z2 ;

…5:17†

which means that there is only one type of topologically stable line disclination. On one hand, each element of the homotopy group corresponds to a class of topologically stable defects; all defects belonging to the same class are equivalent to one another under continuous deformations. On the other hand, the elements of homotopy groups are topological invariants, or topological charges of the defects. The defect-free state corresponds to a unit element of the homotopy group and to a zero topological charge. There is no possibility of transforming a con®guration from one class into a con®guration from another class: transformations 12 $ 0 are prohibited by (in®nitely) high energy barriers. In contrast, transformations 12 ‡ 12 $ 0 are topologically possible; whether they happen or not depends on energies that are of the order of disclination energy or smaller. Let us now consider how the chiral asymmetry changes the classi®cation.

5.5.2

Disclinations in the Cholesteric Phase

Additional interactions, such as dielectric coupling to the electric ®eld, surface anchoring, biaxial or chiral interactions, with energy Fint < Fcond , change the OP space from R to some R at which the sum of energies Fcond ‡ Fint is minimum. R is a submanifold of R [37]. Chiral asymmetry of the molecules leads to the transition from a nematic state with R ˆ S 2 =Z2 which carries all the possible rotations of a unique director, to a cholesteric state with R which carries all the possible rotations of a set of three mutually perpendicular directors. To comply with the terminology of Friedel and KleÂman [41], we will denote these directors l (which shows the local direction de®ned by the molecule), w (along the helical axis), and t ˆ l  w. The OP space is the group G ˆ SO…3† of rotations of the trihedron l, w, t, factored by the four-element point group D2 of p rotations about the directions l, w, and t: R ˆ SO…3†=D2 :

…5:18†

The same result can be obtained for biaxial nematics [42]: from a topological point of view, the classi®cations of defects in cholesterics and biaxial nematics are identical. Calculation of the fundamental group for R ˆ SO…3†=D2 requires knowledge beyond the scope of this chapter. We simply present the result (for details, see [2], [37], [42]): p1 …R† ˆ Q:

…5:19†

Q is the group of quaternion units which consists of eight elements fI ; J; i; ÿi; j; ÿ j; t; ÿtg that obey the multiplication rules:

5. Cholesteric Liquid Crystals: Defects and Topology

133

Figure 5.9. Closed contours (a) G1 and (b), (c) G2 corresponding to jkj ˆ 1 and jkj ˆ 2 disclinations in the OP space of the cholesteric. Both contours connect diametrically opposite and equivalent points at the surface of SO…3†. G1 cannot continuously shrink into a point. G2 runs between the two antipodal points twice (b) and can smoothly leave these points and shrink into a point (c).

i j ˆ ÿ ji ˆ t; JJ ˆ I ;

jt ˆ ÿt j ˆ i; ii ˆ j j ˆ tt ˆ J;

ti ˆ ÿit ˆ j; i jt ˆ J:

…5:20†

The multiplication rules dictate how the disclinations merge, split, and transform. Note that the group operation is noncommutative, i.e., Q is a non-Abelian group. Because of this, each disclination in a cholesteric is characterized not by an element of the fundamental group but by a class of conjugate elements of Q. There are ®ve conjugacy classes: C0 ˆ fI g, C 0 ˆ fJg, Cl ˆ fi; ÿig, Ct ˆ f j; ÿ jg, and Cw ˆ ft; ÿtg. Correspondingly, the topological charge acquires the values 1; ÿ1; …i; ÿi†; … j; ÿ j†, …t; ÿt†. Classes Cl , Ct , and Cw correspond to p rotations of directors when one goes once around the disclination's core; for example, Cl relates to rotations of w and t (l remains nonsingular). Class C 0 corresponds to 2p rotations; unlike their nematic counterparts, these lines are topologically stable. Class C0 describes topologically unstable 4p disclinations. The striking di¨erence between 2p (stable) and 4p (unstable) lines is illustrated in Figure 5.9. The di¨erence between 2p and p (both stable) lines can be illustrated by the following example [43] with w lines (no singularity in the w ®eld). Suppose the w line is perpendicular to equidistant cholesteric layers. The OP space for the nematic director is then S 1 =Z2 , which implies an in®nite number of topologically distinct w lines with integer and half-integer k. When one approaches the core region of the line, the elastic energy @K…`n† 2 increases, until at distances @p it becomes comparable to the energy di¨erence @K=p 2 between the cholesteric and nematic states. At scales smaller than @p, the OP space of the nematic director reverts to S 2 =Z2 . Therefore, the 2p lines with integer k should have a thick core of typical diameter @p that is nonsingular from the nematic point of view: the director is uniform (escaped in third dimension) inside the cylinder of diameter @p. In contrast, p lines with half-integer k are singular both for the uniaxial

134

O.D. Lavrentovich and M. Kleman

Table 5.1. Multiplication rules of ®ve classes of elements of the group Q that control the merger and splitting of disclinations in a cholesteric phase.

C0 C0 Cl Ct Cw

C0

C0

Cl

Ct

Cw

C0 C0 Cl Ct Cw

C0 C0 Cl Ct Cw

Cl Cl C0 or C 0 Cw Ct

Ct Ct Cw C0 or C 0 Cl

Cw Cw Ct Cl C0 or C 0

nematic and cholesteric OPs. Interestingly, the p lines with half-integer k can be further shown to be nonsingular if one allows for biaxiality of the nematic phase and compares the gradient energies at the core to the energy di¨erence between the uniaxial and biaxial states [44]. As a result, in typical thermotropic materials, the core of p lines can be about an order of magnitude wider than the molecular length [44]. The multiplication rules (5.20) are speci®c of the classes of elements, rather than the elements themselves. The results are given in Table 5.1, that can be used to predict the result of merger or splitting of disclinations. If two disclinations from two di¨erent classes merge, the resulting disclination belongs to the class of the product of the ®rst two. The merger of disclinations of the same class from the set Cl , Ct , Cw is ambiguous: the result is either a trivial con®guration (class C0 ) or a disclination from class C 0 , depending on the path of merger with respect to other defect lines in the system [37]. The energy of disclination strongly depends on how the trihedron l, w, t is distorted. In a uniaxial cholesteric, the three have di¨erent physical meaning and di¨erent distortion energy. Only l is a real director while t the w are ``immaterial'' directors; singularities Ct and Cw would be generally more costly than Cl . The di¨erence is seen when the disclinations of half-integer strength k ˆ n ‡ 12 (n is an integer) are compared. The so-called l disclinations in which the director l is not singular are apparently less energetically costly than t disclinations in which l is singular. The core of l disclinations is of radius p (``thick'' lines), Figure 5.10(a), (b), (d), while the core of t disclinations (``thin'' lines) is of molecular size (or somehow larger, as discussed above), Figure 5.10(c). The line tensions thus di¨er by an amount @K ln… p=a†, where a is of the order of 1±10 molecular sizes. If the cholesteric is unwound into a nematic phase, p ! y, then l disclinations vanish. A lÿ (where the superscript ``ÿ'' indicates that the line is of negative strength k ˆ ÿ 12 ) can be annihilated by a collapse with a l‡ , Figure 5.10(b). Figure 5.10(c) pictures a t‡ , i.e., a wedge line of strength k ˆ 12 , singular for the w and l ®elds but continuous for the t ®eld. w lines will be discussed in the next subsection, since they can be treated as dislocations in the system of cholesteric layers.

5. Cholesteric Liquid Crystals: Defects and Topology

135

Figure 5.10. Disclinations l and t in a cholesteric.

The disclinations of integer strength belong to two di¨erent classes: those of odd strength k ˆ 2n ‡ 1 and those of even strength k ˆ 2n. As already mentioned, the disclinations k ˆ 2n ‡ 1 cannot be eliminated; the escape of the director into the third dimension is restricted by a region of size @p. As is clear from Figure 5.10(d), the l line with k ˆ 1 cannot be made continuous for the two ®elds t and w simultaneously. If, say, the t ®eld is made continuous, then the w ®eld remains singular. In contrast, k ˆ 2n lines do escape in the third dimension, Figure 5.9. The distribution of l, t, and w disclinations of di¨erent strength among the ®ve classes of the quaternion group Q is summarized in Table 5.2. The topological classi®cation prohibits transformations of disclinations from one class to another. For example, t…n ‡ 12†, l…n ‡ 12†, and w…n ‡ 12† cannot be continuously transformed one into another, despite the apparent similarity in the value of the ``strength.'' On the other hand, di¨erent lines can be transformed by splitting. For example, according to Table 5.1 and Table 5.2. C0

C0

b ˆ ÿ2np l…2n† t…2n† w…2n†

b ˆ ÿ…2n ‡ 1† p l…2n ‡ 1† t…2n ‡ 1† w…2n ‡ 1†

Cl l…n ‡ 12 †

Ct

Cw b ˆ ÿ…n ‡ 12 † p

t…n ‡ 12 †

w…n ‡ 12 †

136

O.D. Lavrentovich and M. Kleman

(5.20), a Ct line can split into a pair of Cl and Cw lines, if it saves elastic energy. An example of such a splitting (of w lines into lt pairs) will be discussed in the next Section 5.5.3. There is no violation of the law of conservation for the strength k in all these processes since k is not a conserved topological invariant, but just a useful label to distinguish di¨erent con®gurations. One should bear in mind that the topological classi®cation of defects in cholesteric and other layered media such as smectics and ordinary crystals is limited by the condition of the layers equidistance. As a result, some transformations between defects that belong to the same class require very high energy barriers comparable to the energy barriers between di¨erent classes. Transformation l‡ $ lÿ within the class Cl represents such an example. Disclinations l and t are often observed in ®ngerprint textures. Since the line tension of the l lines (with a nonsingular core of size @p) is smaller than the line tension of the singular t lines by an amount @K ln … p=a†, one would expect that l defects are more frequent. However, this analysis might be altered if the cholesteric phase is biaxial: then all three directors might have the same energy weight. Livolant [45] has extensively studied disclinations in the cholesteric textures of three helical biological polymers: DNA, PBG, and xanthan: the l lines were quite frequent, while isolated t lines have never been observed. On the other hand, the t disclinations often appear in pairs with l disclinations to replace w disclinations.

5.5.3

Dislocations

The symmetry of rotations np around the w-axis in cholesterics is equivalent to the symmetry of translations n… p=2†w. Therefore, the w disclinations can be equivalently treated as dislocations [10], [46], with the Burgers vector b ˆ ÿk p:

…5:21†

The values of the Burgers vector are included in Table 5.2. Figure 5.11(a) pictures a w‡ wedge disclination (w is continuous). It can be constructed by a Volterra process performed along the line, by opening the cut surface by an angle p: each cholesteric layer yields a two-dimensional k ˆ 12 con®guration that rotates helically along the line with a pitch p. The equivalence just demonstrated for screw dislocations versus wedge w disclinations can be extended to edge dislocations (Figure 5.12) versus twist w disclinations and even further, to mixed dislocations and disclinations, for the simple reason that the two corresponding Volterra processes are the same. An important property of w dislocations is their ability to split into combinations of l and t disclinations. Of course, these transformations must obey the multiplication rules (5.23). For example, a w line from the class Cw can split into a pair of l and t lines (classes Cl and Ct , respectively). An example is shown in Figure 5.11(b), (c): the core splits into a lÿ and t‡

5. Cholesteric Liquid Crystals: Defects and Topology

137

Figure 5.11. Equivalence in the presentation of w lines: (a) wedge w‡ disclination ˆ screw dislocation; (b) w-twist disclination ˆ edge dislocation; and (c) splitting of the core of a dislocation into a pair of disclinations.

separated by a distance p=4; the Burgers vector is b ˆ p=2, i.e., twice the distance of pairing. Figure 5.12 shows a split dislocation with b ˆ p. Splitting of a dislocation into two disclinations of strength jkj ˆ 12, i.e., of rotation Gp, relates to the fact that the product of two opposite p rotations along two parallel axes W and ÿW …jWj ˆ 1† at a distance d is a translation b ˆ 2W  d:

…5:22†

Therefore, two p disclinations of opposite signs L…W† and L 0 …ÿW† are altogether equivalent to a dislocation L…b†; the ``core'' of the dislocation extends over L and L 0 . Splitting of w disclination lines has been observed in the so-called ``Cano'' wedges: one of the disclinations is always a l, i.e., it does not carry any material singularity. The ®rst dislocations near the center of the wedge have a small Burgers vector …b ˆ p=2†, while b increases for dislocations far from the center. When the cholesteric layers are tilted with respect to the bounding plate, the disclinations might occur to match the twisted structure in the bulk with an (usually unidirectional or conical) orientational ®eld of anchoring forces [47]±[49]. Here again, the w dislocations split into l and t pairs. The l lines

138

O.D. Lavrentovich and M. Kleman

Figure 5.12. Confocal-microscope image of a dislocation b ˆ p in the ``®ngerprint'' cholesteric texture. The confocal microscope technique allows one to obtain the image of the director pattern not only (a) in the plane of the sample but (b), (c) in the vertical cross-section as well (photo D. Voloschenko).

are in the bulk while the t lines are at the surface (which reduces their energy).

5.5.4

Entanglement of Disclinations

One of the most spectacular consequences of noncommutativity of the group Q is the possibility of a topological entanglement of the disclinations. Originally, the problem was considered by Toulouse for biaxial nematics [42], but it applies to any medium with a non-Abelian fundamental group [50]±[52]. Figure 5.13(a) shows two entangled disclinations. The question is whether they can be transformed by continuous variations of the directors into an

5. Cholesteric Liquid Crystals: Defects and Topology

139

Figure 5.13. (a) Entanglement of disclinations in a medium with a trihedron of vectors as the order parameter; (b) topologically trivial; and (c) nontrivial.

Figure 5.14. Continuous deformations of the contour g3 from Figure 5.13 into the ÿ1 product contour g1 g2 gÿ1 1 g2 demonstrating that the image G3 of g3 in OP space is homotopic to the product G1 G2 G1ÿ1 Gÿ1 2 . At step (d), one pinches together four points marked by circles.

140

O.D. Lavrentovich and M. Kleman

unlinked con®guration, Figure 5.13(b), if we require that the ends of the disclinations remain ®xed. To ®nd the answer let us draw three contours g1 , g2 , and g3 from a point M of real space: g1 and g2 encircle the defect lines and g3 encircles the entangled region, Figure 5.13. Their images in OP space will be some contours G1 , G2 , and G3 . Evidently the defects can be unlinked only when G3 is homotopic to zero. If this is not so, then separation of the disclinations will leave a topologically nontrivial trace in space, a third disclination, Figure 5.13(c). The result depends on the nature of the linked disclinations. One can show, Figure 5.14, that the contour G3 is homotopic to the product ÿ1 G1 G2 Gÿ1 1 G2 ; an element of this form is called a commutator in the fundamental homotopy group. For Abelian groups the commutator is the identity element, since G1 G2 ˆ G2 G1 . This is not true for non-Abelian groups; in particular, for the group Q the contour G3 can belong either to the class C0 ÿ1 ÿ1 ÿ1 …G1 G2 Gÿ1 1 G2 ˆ 1† or to the class C 0 …G1 G2 G1 G2 ˆ ÿ1†. The latter situation occurs when the two entangled disclinations belong to di¨erent classes from the set Cl , Ct , Cw . Therefore, after pulling two di¨erent disclinations jkj ˆ 12 across one another, they prove to be connected by a disclination jkj ˆ 1 belonging to C 0 .

5.6

E¨ects of Con®nement

Topological defects are often needed to equilibrate an ordered system. There are two di¨erent possibilities here. First, the defects can occur to relieve intrinsic (``bulk'') frustrations (for instance, between the twist and layered structures in TGB phases). Second, the defects can occur simply because the system is bounded or because there are foreign inclusions, such as colloidal particles or droplets in a liquid±crystalline host. Surface interactions (the phenomenon known as anchoring) change the OP space. One can imagine, for example, a cholesteric bounded by a ¯at wall that imposes strictly normal director orientation. No uniform cholesteric structure of type (5.1) can satisfy this boundary condition; resulting distortions might involve defects. However, this is not the case we have in mind. What we have in mind is that topological defects must appear in the equilibrium state when the bounding surface has a nonzero Euler characteristic. Similar analysis can be performed for particles of di¨erent topology dispersed in the liquid crystal matrix. The condition for this topological consideration to be valid is that the characteristic size of the liquid±crystal system or the dispersed droplet is larger than the anchoring extrapolation length, as will be discussed later. Suppose a two-dimensional vector ®eld n is de®ned on a closed surface with Euler characteristic E. This ®eld might contain point defects whose topological charges are de®ned as

5. Cholesteric Liquid Crystals: Defects and Topology

 ‡ 2 1 1 …2† 1 dn 2 dn n ÿn N 1k ˆ dl ˆ 0; G1; G2; . . . ; 2p dl dl

141

…5:23†

where l is a natural parameter de®ned along the loop enclosing the defect point on the surface. Number k shows how many times n rotates by an angle 2p when one moves around the defect once. The Poincare theorem states that the sum of all charges k of the ®eld n, de®ned at a closed surface, is equal to the Euler characteristic of the surface X kj ˆ E: …5:24† j

For a sphere, E ˆ 2; thus the two point defects at the poles of the nematic droplets in Figure 5.3 illustrate the Poincare theorem; it does not matter if the interior structure is twisted or not. Suppose now that the vector ®eld is three-dimensional. There might be point defects in this ®eld as well. A topological characteristic can be introduced as a number N which counts how many times one meets all possible spatial orientations of the vector ®eld while moving around a closed surface surrounding the point defect. Analytically [53]: n1 n2 n 3  …… qn 1 qn 2 qn 3 ……  1 1 qn qn z qu qu qu du dv ˆ zn  du dv; …5:25† Nˆ 4p 4p qu qv qn 1 qn 2 qn 3 qv qv qv where the coordinates u and v are speci®ed at the surface enclosing the defect. If the vector ®eld is parametrized as n…u; v† ˆ fsin y cos j; sin y sin j; cos yg, where both polar y and azimuthal j angles are functions of u and v, then  ……  1 qy qj qy qj ÿ z sin y du dv: …5:26† Nˆ 4p qu qv qv qu For example, for a radial hedgehog n ˆ r in spherical coordinates …… 1 z sin u du dv ˆ 1: Nˆ 4p

…5:27†

The Gauss theorem states that if the three-dimensional vector ®eld is normal to the closed surface of Euler characteristic E, then the sum of all point defects inside the bounded volume is X Ni ˆ E=2; …5:28† i

i.e., 1 in the case of a sphere. Both the Poincare and Gauss theorems can be applied to structures in cholesteric droplets provided that the surface anchoring is su½ciently strong.

142

O.D. Lavrentovich and M. Kleman

Figure 5.15. Cholesteric textures in spherical droplets with tangential director anchoring at the boundary. Top: A monopole con®guration with a point defect N ˆ 1 in the ®eld w of normals to the cholesteric layers and an attached nonsingular line k ˆ 2, stable when R= p g 1 (microphotograph in crossed polarizers). Bottom: A boojum con®guration with an isolated k ˆ 2 surface point defect at R= p @ 1 (no crossed polarizers). The insert shows the director ®eld at the surface of the droplet.

That is de®nitely the case when the radius R of the droplet is much larger than the anchoring length K=W , where W is the anchoring coe½cient (work per unit area needed to deviate the director from the anchoring direction by an angle, say, 1 rad). The reason is that the typical value of the anchoring energy (resp. the bulk elastic energy) is WR 2 (resp. KR): the surface energy overweighs the bulk elastic energy for large R. If R g p, cholesteric droplets display a monopole-type structure ®rst observed by Robinson and explained by Frank and Price, see [54] and Figures 5.15 and 5.16. The cholesteric layers form a concentric system of spheres. The ®eld w of normals to the layers form a radial point-defect hedgehog with N ˆ 1 in the center, as dictated by (5.28). This point defect cannot be isolated, however: according to (5.24),

5. Cholesteric Liquid Crystals: Defects and Topology

143

Figure 5.16. Con®guration of layers in monopole structures with (a) one k ˆ 2 or (b) two k ˆ 1 disclinations; (c) boojum obtains from the monopole when the line defect shrinks into a point at the boundary.

each cholesteric layer must contain a point defect in the orthogonal ®elds l and t. These point defects form a radial line. There might be one disclination of class C0 with k ˆ 2 or two disclinations of class C 0 with k ˆ 1 [55]±[59]. The overall structure is reminiscent of the Dirac monopole [60], an elementary magnetic charge that carries a point defect of the magnetic ®eld, Bkr, with an attached line singularity in the vector-potential A. The vectorpotential A is normal to B and is thus speci®ed on the concentric spheres around the point, which again brings (5.24) into action and leads to the line singularity in A. The example above shows that the isolated point defects are not likely to occur in the bulk of cholesteric phase when L= p g 1. This is indeed a general statement, valid for any ordered medium, such as super¯uid 3He-A, smectic C, or biaxial nematic with a trihedron of vectors as the order parameter: the second homotopy group for the OP space of these media is trivial. However, point defects at the boundary of the cholesteric volumes and all the media listed above are formally allowed by the homotopy theory. The point defect at a surface of an ordered medium can represent either the end of a line that is topologically stable in the bulk or a true surface point defect with no bulk singularity attached [61]. In cholesteric liquid crystals, all points with jkj ˆ 12 ; 1 are the ends of bulk disclinations. Only when jkj ˆ 2 (4p rotations of the director ®eld), the point defect might be an isolated surface singularity. However, even in this case one should take care of the requirement of the layers equidistance. For example, the classical boojum con®guration cannot be observed in a cholesteric vessel when L= p g 1. The boojum has been introduced by Mermin [62] for super¯uid 3He-A as a way of reducing the energy of the monopole. In the 3He-A spherical volume, the energy of a monopole decreases when the line shrinks into a point at the surface; this point is the boojum. However, in the cholesteric phase, such a transformation violates the equidistance between the layers. As a result, the monopole structure remains stable, at least when R=p g 1. Only when R= p @ 1, can the isolated point defect with 4p rotations of the director ®eld

144

O.D. Lavrentovich and M. Kleman

be observed. Figure 5.15 shows how the cholesteric monopole is replaced by a boojum when the pitch increases [6], [55]. Cholesteric droplets have been extensively studied during the last decade, especially after Crooker and Yang suggested to use polymer-dispersed cholesteric liquid crystals for re¯ective color displays [63]. Lavrentovich and Nastishin [64], [65] reported on an intriguing phenomenon: liquid crystal droplets dispersed in an isotropic matrix (glycerin with lecithin) divided into smaller ones when one decreases the temperature of the sample, and passes from the cholesteric to the smectic A phase through the TGB phase. The reader is referred to the recent reviews [66]±[68], and to the contribution of Crawford, SvensÏek, and Zumer in this book for more details about dispersed liquid crystals. Much less is known about the inverted systems, in which the liquid crystal serves as a host to foreign particles, say, spherical silica particles or water droplets. The topological consideration above can be applied to these systems as well, as soon as the dispersed particles are large enough for the surface anchoring to set a well-de®ned director orientation at the particle surface. Since each spherical particle is a seed of a nonzero Euler characteristic …E ˆ 2†, the topological defects are intrinsic to these systems and de®ne many important physical features, such as stability against coagulations observed for isotropic droplets in the nematic host [69].

5.7

Nonsingular Con®gurations and Solitons

The concept of OP space helps us to analyze complicated con®gurations of the cholesteric order parameter even when these con®gurations are topologically trivial, i.e., equivalent to an undistorted cholesteric or nematic. Suppose the cholesteric is con®ned between two homeotropic plates separated by a distance L @ p. The magnetic ®eld acts along the normal h to the plates. If the ®eld is su½ciently strong, or the sample is su½ciently thin, the cholesteric is in the homeotropic nematic state with the director n ˆ Gh; the OP space is reduced to a single point. When the ®eld decreases, the balance of diamagnetic, elastic, and surface anchoring energies result in complicated con®gurations such as ``spherulites'' and ``®ngers'' [70]±[74]. Inside, the cholesteric twists; at the boundary of the con®guration, the director adopts a homeotropic orientation in order to match the surrounding matrix, Figure 5.17(a). Usually, this twist is nonsingular (although some types of con®gurations might contain line [75] or point [76] singularities). If L= p @ 1, a convenient way to analyze both the geometry and energetic stability of con®gurations is to map the director ®eld onto the sphere S 2 [71]. For example, the double-twisted director ®eld of the ®nger in Figure 5.17(a) is represented by a lobe on S 2 in Figure 5.17(b). Clearly, the lobe can shrink into a point, say, the north pole of S 2 ; thus the ®nger is equivalent to the uniform state n ˆ Gh. Any other con®guration (even with topological singularities such as

5. Cholesteric Liquid Crystals: Defects and Topology

145

(a)

(b) Figure 5.17 Computer-generated cross-section of a cholesteric ``®nger'' in a cell with ®nite surface anchoring and magnetic ®eld: director con®guration (a) in the real space and (b) on the sphere S 2 (courtesy S.V. Shiyanovskii).

146

O.D. Lavrentovich and M. Kleman

Figure 5.18. Twist wall as a planar soliton; director ®eld in (a) the real space and (b) the OP space.

a pair of radial and hyperbolic hedgehogs described by Pirkl et al. [76] for spherulites) that appears from the homeotropic state, should have an image on S 2 that is shrinkable into a point. Of course, when the ®eld is weak, the gain in the twist energy prevents the lobe from shrinking. Fingers, spherulites, and core structures of the disclinations considered in Section 5.2 have the following common feature: the OP space looks di¨erent at di¨erent scales. This feature is characteristic of topological solitons that are described by relative homotopy groups [77], [43]. To illustrate the issue, consider ®rst a twisted wall in a nematic cell with tangential boundary conditions subjected to a strong horizontal magnetic ®eld, Figure 5.18(a). The wall might terminate on a disclination line, or, if the line is moved to the sample's boundary on the left, run along the entire sample. Consider the mapping of the line gn threaded through the wall into the OP space. The ends of the line are mapped into antipodal identical points n ˆ Gh, while the line gn itself is mapped onto the closed contour Gn linking these points in the OP space. This contour cannot be contracted to a point by any continuous transformations, as long as the end points are ®xed. Therefore, the deformed (but nonsingular) con®guration is stable. The width of the wall is ®xed by the balance of elastic, anchoring, and ®eld energies. Such a structure is called a topological soliton. A soliton of the planar type just described can be closed into a loop or terminate at disclinations. Alternatively, a disclination loop (or a pair of disclinations) can nucleate in the plane of the soliton and destroy the wall. In the general case, the classes of homotopic mappings of the line g threaded through a planar soliton form the relative homotopy group p1 …R; R†, where R is the OP space far from the core of the soliton, shrunk (as compared to the complete OP space R) by additional interactions (external ®eld, boundary conditions, etc.). If R consists of a single point, as in Figure 5.18, p1 …R; R† coincides with the fundamental group p1 …R† [77], [78]. Just as a disclination in an external ®eld can give rise to a planar soliton, a point defect can give rise to a linear soliton. Linear solitons are described by the classes of mappings of the surface s crossing the soliton into the OP spaces R and R, i.e., by the elements of the second relative group p2 …R; R†.

5. Cholesteric Liquid Crystals: Defects and Topology

147

As already discussed, there are no isolated point defects in the cholesteric phase, p2 …R ˆ SO…3†=D2 † ˆ 0. However, singular point defects can serve as the ends of linear solitons, as in the case of the monopole structure, in which the nonsingular disclinations can be considered as linear solitons. The most interesting case is presented by the so-called particle solitons. The distribution of the OP in particle-like solitons depends on all three coordinates. They are described by the group p3 …R; R† of homotopy classes of the mappings of the three-dimensional spherical volume D 3 containing the soliton into the OP space R. Here the boundary of the spherical volume, the sphere s, is mapped into the shrunk space R. If R consists of one point, then the particle-like soliton is described by the group p3 …R†. The spherical volume D 3 with all points of its surface s being equivalent, is homotopic to a three-dimensional sphere S 3 in a four-dimensional space. Thus the elements of p3 …R† are the mappings S 3 ! R. The special cases S 3 ! S 2 and S 3 ! S 2 =Z2 are called Hopf mappings, Figure 5.19, and correspond to p3 …S 2 † ˆ p3 …S 2 =Z2 † ˆ p3 …SO…3†=D2 † ˆ Z, where Z is the group of integers; classi®cation for the cholesteric and nematic phases is the same. In a uniaxial nematic, the particle-like soliton amounts to a director con®guration distorted in a region of ®nite size, outside of which the director ®eld is uniform. As a rule, such solitons are unstable with respect to a decrease in size and subsequent disappearance on scales smaller than the

Figure 5.19. A nontrivial Hopf texture in a three-dimensional vector ®eld, as seen in the vertical cross-section. The vector ®eld is directed north everywhere outside the sphere and at the origin. The vertical axis is the rotational symmetry axis. When going along any radius from the center to the surface of the sphere, the vector rotates by an angle 2pr=R around this radius. The length of the arrows is proportional to the length of the vector projection in the XY plane.

148

O.D. Lavrentovich and M. Kleman

coherence length x. The decrease in size L ! Lm …m < 1† entails an increase in the elastic-energy density by a factor of 1=m 2 and a decrease in the volume of the soliton by a factor of m 3 , so that the total elastic energy decreases: F ! F m. Stabilization of particle-like solitons can be facilitated by an additional interaction, in particular, by helical twisting of the structure [79]. In a weakly twisted cholesteric mixture Bouligand [80], [81] observed two linked disclination rings k1 ˆ k2 ˆ 1, each of which, by itself, is topologically unstable, whereby all points of the cores of the disclination are mapped into a single point of S 2 =Z2 . In going from one ring to the other, the director undergoes a p rotation and one can represent the rings as inverse images of two diametrically opposite points on S 2 . Evidently one cannot convert the con®guration into a homogeneous state because the rings are linked: upon trying to unlink the rings, they must intersect one another and singularities would arise in the con®guration. The degree of linking of the rings, equal in this case to unity, coincides with the Hopf invariant, which is an element of the group p3 …S 2 =Z2 † ˆ Z. The stability of the con®guration as a whole is guaranteed by the conservation of the Hopf invariant [81].

5.8

E¨ects of the Layer Structure

As already mentioned, the layered structure of cholesteric materials imposes certain limitations on the topological classi®cation of defects based on homotopy groups; a more general theory is still lacking. In this section we discuss macroscopic defects such as focal conic domains and oily streaks whose existence depends crucially on the layered character of ordering.

5.8.1

Focal Conic Domains

In the regime L=p g 1, the elastic theory considers the cholesteric medium as a system of equidistant (and thus parallel) layers and that the curvature distortions are predominant, (5.8). The description of defects such as edge dislocations, oily streaks, and focal conic domains in cholesterics is often based on the results obtained for ``simpler'' layered medium, namely, the smectic A phase. The liquid crystal samples are always bounded, so the surface interactions prescribe a certain orientation of the cholesteric layers (most often, the layers align parallel to the bounding surface). Generally, the two requirements (surface orientation and the equidistance of layers) can be satis®ed simultaneously only when the layers are bent in a very special manner. As was established by G. Friedel and Grandjean, originally for smectic A phases [82], [83], all the parallel layers should take the shape of ``Dupin cyclides.'' According to Dupin (see, e.g., Darboux [84]), in such a case the two focal surfaces of the layers are degenerate into lines, which are confocal conics (e.g., an ellipse and a hyperbola; or a circle and a straight line; or two pa-

5. Cholesteric Liquid Crystals: Defects and Topology

149

Figure 5.20. Focal conic domain with a circular base smoothly embedded in the set of horizontal ¯at layers. The layers are equidistant everywhere except in the vicinity of the circle and the confocal straight line.

rabolas). The restricted part of space ®lled with a single family of Dupin cyclides is called a focal conic domain (FCD). Figure 5.20 shows a particular type of toroidal FCDÐa circle and a straight line as focal lines (for more details, see [2], [85]). The layered structure inside the FCD exactly matches the planar con®guration of the layers outside. Although the FCDs involve three-dimensional distortions, the elastic energy of an FCD scales linearly with the characteristic size @KL (e.g., the size of the elliptical or circular base). When an FCD has its base on the bounding surface, it e¨ectively changes the surface orientation of cholesteric layers: in most experimental situations, the layers are perpendicular to the boundary inside the base, while outside they are parallel. When the surface does not favor the planar orientation of layers, the appearance of FCDs with size L > L  ˆ K=…gk ÿ g? † is energetically justi®ed [86]: every domain saves surface energy …gk ÿ g? †L 2 at the expense of elastic energy KL. Recent atomic force examination of FCDs in cholesteric oligomers by Meister et al. [87] reveals smooth matching between layers tilted at the free surface of the sample (inside the FCDs) and ¯at layers in the bulk that are parallel to the interface (outside the FCDs). Focal conic textures and their transformations under applied electric ®eld, studied for smectic layered systems [88], [89], are used in bistable cholesteric re¯ective displays [90]. The SMLC cholesterics most frequently present polygonal textures with domains of a negative Gaussian curvature. In these domains the focal conditions are not exactly satis®ed [91], and unlike the situation in smectics, the cholesteric layers might deviate from the exact geometry of Dupin cyclides

150

O.D. Lavrentovich and M. Kleman

by changing their thickness. In a cholesteric, it is possible to let the pitch vary by a large range at the sole expense of some energy of twist K2 . These polygonal textures are less frequent in biopolymers in solution (DNA, polypeptides, etc.) [45], because the twist energy is accompanied by a considerable bend contribution K3 , caused by the rigidity of the long molecules. We are not aware of any reports on FCDs with a positive Gaussian curvature in the bulk of cholesteric materials (although such domains are documented for lyotropic lamellar phases [92]). The most rigid cholesteric biopolymers have other types of layer textures, the monopole structures or Robinson spherulites discussed in the previous section: the layers are approximately along concentric spheres (positive Gaussian curvature). An extensive study and review of cholesteric spherulites in materials of biological interest can be found in Bouligand and Livolant [57].

5.8.2

Oily Streaks

Oily streaks and liquid crystals were discovered simultaneously. In 1888, F. Reinitzer studied cholesterylbenzoate and noticed elongated ``¯uid'' inclusions in the cholesteric sample [93]. Oily streaks, as FCDs, are common for many lamellar liquid crystalls. In a ¯at cell with layers parallel to the bounding plates, oily streaks appear as long bands that divide the ideal domains of ¯at layers. Their inner structure is quite complicated and depends on many parameters, most notably on elastic constants [94] (including the saddle-splay elastic constant [95]) and surface anchoring. According to Friedel [83], oily streaks are made of pairs of edge dislocations of (large) opposite Burgers vectors nd, n 0 d, making a total Burgers vector b ˆ …n ÿ n 0 †d; here d is the characteristic interlamellae distance, such as the thickness of a smectic A layer or the half-pitch in cholesterics. Each element of the pair is most probably due to the coalescence of small Burger's vector dislocations of the same sign. A large Burgers vector dislocation b ˆ nd can be favoured with respect to small Burgers vectors dislocations ni d, P i ni ˆ n, produced when the sample is formed. The explanation is in the speci®c model of the dislocation core which is split into two disclinations of opposite signs, Figure 5.21(a); the splitting reduces the total elastic energy of the dislocation [94]. A further feature characteristic of oily streaks is the frequent occurrence of a transversal striation caused either by undulation of the layers or by the formation of FCD chains [95]. The simplest variety of oily streaks is shown in Figure 5.21(b): two parallel k ˆ 12 disclinations with a wall between them. The total Burgers vector is zero, so that the oily streak is topologically trivial and can disappear by pulling the semiround ends together. There is no transversal striation so that the Gaussian curvature is zero everywhere except at the end region (where it is negative). Since the line tension (the free energy per unit length) of the oily streaks is

5. Cholesteric Liquid Crystals: Defects and Topology

151

(a)

(b) Figure 5.21. (a) Splitting of the core of a large Burgers vector dislocation into two disclinations; and (b) an oily streak with a semicircular end in a system of parallel and equidistant layers.

normally positive, their networks coarsen with time. Zapotocky et al. [96] suggested stabilizing the networks of oily streaks by adding micron-size colloidal particles to the cholesteric. The particles gather at the nodes of the network. The stabilized network of connected oily streaks greatly modi®es the rheological properties of the system, making it gel-like. In contrast to a

152

O.D. Lavrentovich and M. Kleman

Figure 5.22. Network of oily streaks in a cholesteric planar texture. Cell thickness 15 mm, cholesteric pitch 0.48 mm. The oily streaks are (a) straight at zero voltage but (b) buckle when the voltage exceeds some threshold value. The short side of the ®eld of view is 420 mm. The narrow streaks do not buckle while the wide streaks start to buckle at voltages higher than the medium width streaks.

defect-free cholesteric ¯uid that exhibits liquid-like rheology, the stabilized oily streaks exhibit macroscopic rubber-like elasticity [96]. Usually, the layers within the FCDs and oily streaks, and the layers outside these defects, have di¨erent orientation at the sample's boundaries; thus the problems of surface anchoring and the layers' curvature in these defects are strongly connected. To illustrate the relationship, we brie¯y discuss an electric ®eld instability that manifests itself as a buckling of oily streaks [97]. The electric ®eld changes the line tension of oily streaks and can even drive it negative, in which case the oily streaks buckle and proliferate rather than coarsen. Figure 5.22(a) shows the network of connected oily streaks in a cholesteric sample when the electric ®eld is absent. The bounding plates of the sample are treated to align the molecules parallel to the plates, so that the ®eld w is normal to the bounding plates. The edges that separate the uniform domains are oily streaks, provoked by inhomogeneities such as plastic spacers that keep the binding glass plates apart. The width 2a of the oily streak is de®ned by the number of layers that undergo a p turn. An electric ®eld E is applied to transparent conducting ITO layers at the bounding plates and is thus normal to the cholesteric layers at the faces. The dielectric anisotropy of the cholesteric material (de®ned with respect to the helix axis w) is negative, ea < 0, so that the layers tend to reorient along E. One would expect expansion of the oily streaks since the layers are almost parallel to E inside the oily streak. The experiment, Figure 5.22(b), shows that the expansion takes place as an elongation and buckling of the streaks rather than as their widening: the ®eld drives the line tension of oily streaks negative.

5. Cholesteric Liquid Crystals: Defects and Topology

153

Although the phenomenon is similar in appearance to the buckling of dislocations in Cano wedges [98], [99], it cannot be understood solely as a dielectric response [99] of the cholesteric. The behavior of the oily streaks can be explained only if one adds a speci®c surface anchoring term with an amplitude W @ K= p to the standard energy density, comprised of the elastic terms (5.7) and the dielectric term @ ÿea …w
E† 2 . The anchoring term is caused by tilting the cholesteric layers at the boundaries. The tilt angle increases from 0 outside the streak to ymax ˆ arctan x at the vertical middle plane of the streak, see Figure 5.21(b); x ˆ 2a=h is the width of the streak renormalized by the cell thickness h. The anchoring penalty increases when 2a increases (since ymax increases); this increase can be avoided if the oily streak simply elongates, preserving ymax ˆ const. A quantitative analysis [97] with the model of the oily streak depicted in Figure 5.21(b), shows that the elastic, dielectric, and anchoring contributions to the line tension F of the oily streak depend di¨erently on x. The elastic term is practically linear with x (the slowly changing logarithmic factor @ ln x can be neglected). The dielectric and anchoring energies scale as x 2 when x f 1, but switch to a linear scaling @ x when x g 1. Thus narrow oily streaks are always dominated by the elastic energy and F > 0 for any applied voltage. Anchoring takes over at x g 1, so that the line tension of the wide streaks is also positive. For intermediate x @ 1, when the ®eld is higher than some threshold value Vth , the (negative) dielectric contribution outbalances both the elastic and anchoring terms and drives the line tension negative. The oily streak elongates, preserving the width that corresponds to the minimum of the curve F …x† in Figure 5.23, as in the experiment, Figure 5.22(b).

Figure 5.23. Line tension of the oily streak as a function of its width for di¨erent values of the applied ®eld. Note that at high ®eld only the streaks of intermediate width gain a negative line tension.

154

O.D. Lavrentovich and M. Kleman

Note that a good quantitative agreement between the model and experiment can be obtained only when there is an ``intrinsic'' contribution W @ K= p to the anchoring coe½cient. It comes from the layered structure of the cholesteric and has no analog in the nematic phase (but has a counterpart in smectics). The estimate W @ K= p is natural since the number of cholesteric layers crossing the boundary is of the order of y=p, and each layer has to be distorted (continuously or through dislocations) to accommodate the anchoring direction at the boundary. The energy of these distortions is @ K per layer. Hence W @ K= p.

5.9

Conclusion

The molecular chirality and tendency of many liquid crystalline phases (even composed of nonchiral molecules) to twist results in spectacular director con®gurations that often include topological defects and solitons. We covered the very basic properties of deformations related to twisted structures; the considerations were restricted mainly to a uniaxial cholesteric phase in a static regime. Some of the areas where one can expect further exciting progress are listed below. Although the homotopy theory of classifying defects in condensed media had been developed more that 20 years ago, some theoretical aspects, such as direct inclusion of the large-scale equidistance of layers, can still be advanced. Another ®eld awaiting both theoretical and experimental e¨ort is the hydrodynamics of chiral systems, especially in the presence of defects. Experimentally, some of the predictions of homotopy theory, such as the topological entanglement of disclinations in media with a director's trihedron as the order parameter, remain to be observed. Experimental studies of phases, in which the frustrations between layered (smectic-like) or hexagonal order and twist (or double-twist) deformations are resolved through topological defects, are still in their infancy; major breakthroughs are expected for new liquid crystal materials composed of discotic molecules and polymers. Many cholesteric structures, especially in the regime L= p @ 1, are too sophisticated to be reconstructed by analytical analysis or by ordinary polarizing microscopy. Development of computer simulations and relatively new experimental techniques, such as confocal microscopy, promises fast progress in the deciphering of chiral con®gurations. With this background information, the research can further progress into the fascinating areas of new complex materials, in which the liquid crystal serves as the dispersed phase (as in the polymer-dispersed liquid crystals [67], [68]) or as the dispersion medium, as in the example [96] with defect-stabilized cholesteric gels. Defects are topologically inherent in all liquid±crystal disperse systems that have internal surfaces, as discussed in Section 5.6. One could de®nitely expect a strengthening of the interplay between cholesteric liquid crystal and biological studies: chirality is one of the most

5. Cholesteric Liquid Crystals: Defects and Topology

155

important features of living species. On a more immediate pragmatic side, studies of defects in cholesteric cells subjected to external ®elds will advance the design of re¯ective displays [90] and other devices such as ``smart mirrors'' [100], or di¨raction gratings with electrically controlled periodicity [101]. Acknowledgment. This work was supported by the NSF US±France Cooperative Scienti®c Program, Grant No. INT-9726802 and by NSF STC ALCOM under Grant No. DMR89-20147.

References [1] P.G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed., Oxford Science, Clarendon Press, Oxford, 1993. [2] M. KleÂman, Points, Lines, and Walls, Wiley, Chichester 1983. [3] P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, 1995; T.C. Lubensky, Solid State Commun. 102, 187 (1997). [4] D.R. Link, G. Natale, R. Shao, J.E. Maclennan, N.A. Clark, E. KoÈrblova, and D.M. Walba, Science 278, 1924 (1997). [5] G. Heppke and D. Moro, Science 279, 1872 (1998). [6] M.V. Kurik and O.D. Lavrentovich, Pis'ma Zh. Eksp. Teor. Fiz. 35, 362 (1982), [JETP Lett. 35, 444 (1982)]. [7] L.M. Blinov and V.G. Chigrinov, Electrooptic E¨ects in Liquid Crystal Materials, Springer-Verlag, New York, 1994. [8] G.P. Crawford and S. Zumer, in: Liquid Crystals in the Nineties and Beyond (edited by S. Kumar), p. 331, World Scienti®c, Singapore, 1995. [9] R.B. Meyer, in: Polymer Liquid Crystals (edited by A. Ci¨eri et al.), Ch. 6, Academic, New York, 1982. [10] M. KleÂman, Rep. Prog. Phys. 52, 555 (1989). [11] C. Mauguin, Bull. Soc. France Mineral 34, 71 (1911). [12] D. Meyerhofer, A. Sussman, and R. Williams, J. Appl. Phys. 43, 3685 (1972). [13] O.D. Lavrentovich, Phys. Rev. A 46, R722 (1992). [14] G. Srayer, F. Lonberg, and R.B. Meyer, Phys. Rev. Lett. 67, 1102 (1991); R.B. Meyer, F. Lonberg, V. Taratuta, S. Fraden, S.-D. Lee, and A.J. Hurd, Faraday Disc. Chem. Soc. 79, 125 (1985). [15] M.J. Press and A.S. Arrott, Phys. Rev. Lett. 33, 403 (1974). [16] F. Lonberg and R.B. Meyer, Phys. Rev. Lett. 55, 718 (1985). [17] G.E. Volovik and O.D. Lavrentovich, Zh. Eksper. Teor. Fiz. 85, 1997 (1983), [Sov. Phys. JETP 58, 1159 (1983)]. [18] R.D. Williams, J. Phys. A: Math. Gen. 19, 3211 (1986). [19] M. KleÂman, J. Phys. (Paris) 46, 1193 (1985). [20] M. KleÂman, J. Phys. Lett. 46, L-723 (1985). [21] D.C. Wright and N.D. Mermin, Rev. Mod. Phys. 61, 385 (1989). [22] F. Livolant and Y. Bouligand, Chromosoma 80, 97 (1980). [23] J. Friedel, Proc. 6th Gen. Conf. Eur. Phys. Soc. (Prague) (edited by J. Santra and J. Panto¯iceÂk). [24] M. KleÂman, Phys. Scripta T19, 565 (1987).

156

O.D. Lavrentovich and M. Kleman

[25] F. Livolant and Y. Bouligand, J. Phys. (Paris) 47, 1813 (1986); F. Livolant, A.M. Levelut, J. Doucet, and J.P. Benoit, Nature 339, 724 (1989). [26] D.H. Van Winkle, M.W. Davidson, W.-X. Chen, and R.L. Rill, Macromolecules 23, 4140 (1990); D.C. Martin and E.L. Thomas, Phil. Mag. A 64, 903 (1991). [27] R.D. Kamien and D.R. Nelson, Phys. Rev. Lett. 74, 2499 (1995); Phys. Rev. E53, 650 (1996). [28] T.C. Lubensky, A.B. Harris, R.D. Kamien, and G. Yan, Ferroelectrics 212, 1 (1998). [29] P.G. de Gennes, Solid State Comm. 10, 753 (1972). [30] S.R. Renn and T.C. Lubensky, Phys. Rev. A 38, 2132 (1988); J. Goodby, M.A. Waugh, S.M. Stein, E. Chin, R. Pindak, and J.S. Patel, Nature 337, 449 (1988); T.C. Lubensky and S.R. Renn, Phys. Rev. A 41, 4392 (1990); S.R. Renn, Phys. Rev. A 45, 953 (1992). [31] B.A. Dubrovin, S.P. Novikov, and A.T. Fomenko, Sovremennaya Geometriya, Moscow, Nauka, 760 pp. (1979) (in Russian); [English transl.: Modern Geometry, Parts I, II, III, Springer-Verlag, 1985±1992]; A.Yu. Grosberg and A.R. Khokhlov, Statistical Physics of Macromolecules, AIP Press, New York, 1994. [32] F.B. Fuller, Proc. Nat. Acad. Sci. USA 68, 815 (1981). [33] D. Swigon, B.D. Coleman, and I. Tobias, Biophys. J. 74, 2515 (1998). [34] DNA Topology and its Biological E¨ects (edited by N. Gozzarelli and J.C. Wang), Cold Spring Harbor Laboratory Press, 1990. [35] G. Toulouse and M. KleÂman, J. Phys. Lett. (Paris) 37, L-149 (1976); M. KleÂman, J. Phys. Lett. (Paris) 38, L-199 (1977). [36] G.E. Volovik and V.P. Mineev, Pis'ma Zh. Eksp. Teor. Fiz. 24, 605 (1976); [JETP Lett. 24, 595 (1976)]. [37] G.E. Volovik and V.P. Mineev, Zh. Eksper. Teor. Fiz. 72, 2256 (1977); [Sov. Phys. JETP 45, 1186 (1977)]. [38] M. KleÂman and L. Michel, Phys. Rev. Lett. 40, 1387 (1978). [39] P. Cladis and M. KleÂman, J. Phys. (Paris) 33, 591 (1972). [40] R.B. Meyer, Phil. Mag. 27, 405 (1973). [41] J. Friedel and M. KleÂman, Fundamental Aspects of Dislocation Theory (edited by J.A. Simmons, R. de Wit, and R. Bullough), Special Publication No. 317, Washington, DC: National Bureau of Standards 1, 715 (1970). [42] G. Toulouse, J. Phys. Lett. 38, L67 (1977). [43] V.P. Mineev, Soviet Scienti®c Reviews, Section A: Physics Reviews, vol. 2 (edited by I.M. Khalatnikov), p. 173, Harwood Academic, London, 1980. [44] I.F. Lyuksyutov, Sov. Phys. JETP 75, 358 (1978). [45] F. Livolant, J. Phys. 47, 1605 (1986). [46] Y. Bouligand and M. KleÂman, J. Phys. France 31, 1041 (1970). [47] P.E. Cladis and M. KleÂman, Mol. Cryst. Liq. Cryst. 16, 1 (1972). [48] A. Saupe, Mol. Cryst. Liq. Cryst. 21, 211 (1973). [49] R. Meister, H. Dumoulin, M.-A. HalleÂ, and P. Pieranski, J. Phys. II France 6, 827 (1996). [50] V. Poenary and G. Toulouse, J. Phys. (Paris) 38, 887 (1977). [51] M.I. Monastyrsky and V.S. Retakh, Commun. Math. Phys. 103, 445 (1986). [52] K. Janich and H.R. Trebin, Physique des DeÂfauts/Physics of Defects (edited by R. Balian et al.), p. 421, Les Houches, Session XXXV, 1980; North-Holland, Amsterdam, 1981.

5. Cholesteric Liquid Crystals: Defects and Topology

157

[53] M. KleÂman, Phil. Mag. 27, 1057 (1973). [54] C. Robinson, Trans. Faraday Soc. 52, 571 (1956); C. Robinson, J.C. Ward, and R.B. Beevers, Discuss. Faraday Soc. 25, 29 (1958). [55] M.V. Kurik and O.D. Lavrentovich, Mol. Cryst. Liq. Cryst. (Lett.) 72, 239 (1982). [56] M.V. Kurik and O.D. Lavrentovich, Zh. Eksper. Teor. Fiz. 85, 511 (1983); [Sov. Phys. JETP 58, 299 (1983)]. [57] Y. Bouligand and F. Livolant, J. Phys. (Paris) 45, 1899 (1984). [58] J. Bezic and S. Zumer, Liq. Cryst. 11, 593 (1992). [59] F. Xu and P.P. Crooker, Phys. Rev. E56, 6853 (1997); R.R. Swisher, H. Hue, and P.P. Crooker, Liq. Cryst. 26, 57 (1999). [60] P.A.M. Dirac, Proc. Roy. Soc. London, A 133, 60 (1931). [61] G.E. Volovik, Pis'ma Zh. Eksp. Teor. Fiz. 28, 65 (1978); [JETP Lett. 28, 59 (1978)]. [62] N.D. Mermin, in: Quantum Fluids and Solids (edited by S.B. Trickey, E.D. Adams, and J.F. Dufty), p. 3, Plenum Press, New York, 1977, Boojums All the Way Through, Cambridge University Press, New York, 1990, 310 pp. [63] P.P. Crooker and D.K. Yang, Appl. Phys. Lett. 57, 2529 (1990). [64] O.D. Lavrentovich and Yu.A. Nastishin, Pis'ma Zh. ETP 40, 242 (1984); [Sov. Phys. JETP Lett. 40, 1015 (1984). [65] O.D. Lavrentovich, Yu.A. Nastishin, V.I. Kulishov, Yu.S. Narkevich, A.S. Tolochko, and S.V. Shiyanovskii, Europhys. Lett. 13, 313 (1990). [66] H.-S. Kitzerow, Liq. Cryst. 16, 1 (1994). [67] P.S. Drzaic, Liquid Crystal Dispersions, World Scienti®c, Singapore, 1995, 430 pp. [68] O.D. Lavrentovich, Liq. Cryst. 24, 117 (1998). [69] P. Poulin, H. Stark, T.C. Lubensky, and D.A. Weitz, Science 275, 1770 (1997). [70] M. Brehm, H. Finkelmann, and H. Stegemeyer, Ber. Bunsen-Ges. Phys. Chem. 78, 883 (1974). [71] F. Lequeux, P. Oswald, and J. Bechhoefer, Phys. Rev. A40, 3974 (1989); F. Lequeux, J. Phys. France 49, 967 (1988). [72] J. Baudry, S. Pirkl, and P. Oswald, Phys. Rev. E57, 3038 (1998). [73] L. Gil and G.M. Gilli, Phys. Rev. Lett. 80, 5742 (1998). [74] T. Nagaya, Y. Hikita, H. Orihara, and Y. Ishibashi, J. Phys. Soc. Jap. 67, 2546 (1998). [75] P. RibieÁre, P. Oswald, and S. Pirkl, J. Phys. II France 4, 127 (1994). [76] S. Pirkl, P. RibieÁre, and P. Oswald, Liq. Cryst. 13, 413 (1993). [77] V.P. Mineyev and G.E. Volovik, Phys. Rev. B 18, 3197 (1978). [78] R. Kutka, H.-R. Trebin, and M. Kiemes, J. Phys. France 50, 861 (1989). [79] R.D. Pisarski and D.L. Stein, J. Phys. (Paris) 41, 345 (1980). [80] Y. Bouligand, J. Phys. France 35, 959 (1974). [81] Y. Bouligand, B. Derrida, V. Poenaru, Y. Pomeau, and G. Toulouse, J. Phys. France 39, 863 (1978). [82] G. Friedel and F. Grandjean, Bull. Soc. Franc. MineÂr. 33, 192 and 409 (1910); C. R. Hebd. SeÂan. Acad. Sci. 151, 762 (1910). [83] G. Friedel, Ann. Phys. (Paris) 18, 237 (1922). [84] G. Darboux, TheÂorie GeÂneÂrale des Surfaces, republished by Chelsea, New York, 1954.

158

O.D. Lavrentovich and M. Kleman

[85] P. Boltenhagen, M. KleÂman, and O.D. Lavrentovich, in: Soft Order in Physical Systems (edited by Y. Rabin and R. Bruinsma), p. 5, Plenum Press, New York, 1994. [86] O.D. Lavrentovich, Zh. Eksp. Teor. Fiz. 91, 1666 (1986); [Sov. Phys. JETP 64, 984 (1986)]. [87] R. Meister, M.-A. HalleÂ, H. Dumoulin, and P. Pieranski, Phys. Rev. E 54, 3771 (1996). [88] O.D. Lavrentovich, M. KleÂman, and V.M. Pergamenschik, J. Phys. II (Paris) 4, 377 (1994). [89] Z. Li and O.D. Lavrentovich, Phys. Rev. Lett. 73, 280 (1994). [90] D.K. Yang and J.W. Doane, SID Technical Papers Digest 23, 759 (1992). [91] Y. Bouligand, J. Phys. (Paris) 33, 715 (1972). [92] P. Boltenhagen, O.D. Lavrentovich, and M. KleÂman, Phys. Rev. A 46, R1743 (1992); P. Boltenhagen, M. KleÂman, and O.D. Lavrentovich, C. R. Acad. Sci. Paris 315, SeÂrie II, 931 (1992). [93] F. Reinitzer, Monatsh. Chem. 9, 421 (1888). [94] M. KleÂman and C.E. Williams, Phil. Mag. 28, 725 (1973); J. Phys. Lett. 35, L-49 (1974). [95] P. Boltenhagen, O.D. Lavrentovich, and M. KleÂman, J. Phys. II France 1, 1233 (1991). [96] M. Zapotocky, L. Ramos, P. Poulin, T.C. Lubensky, and D.A. Weitz, Science 283, 209 (1999). [97] O.D. Lavrentovich and D.-K. Yang, Phys. Rev. E57, R6269 (1998). [98] Orsay liquid crystal group, Phys. Lett. A 28, 687 (1969). [99] M. KleÂman and J. Friedel, J. Phys. (France) 30, C4-43 (1969). [100] R.B. Meyer, F. Lonberg, and C.-C. Chang, Mol. Cryst. Liq. Cryst. 288, 47 (1996). [101] D. Subacius, S.V. Shiyanovskii, Ph. Bos, and O.D. Lavrentovich, Appl. Phys. Lett. 71, 3323 (1997).

6

Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics Guram Chilaya

Cholesteric liquid crystals (CLCs) show very distinctly that molecular structure and external ®elds have a profound e¨ect on cooperative behavior and phase structure (see also Chapters 2 and 3). CLCs possess a supermolecular periodic helical structure due to the chirality of molecules. The spatial periodicity (helical pitch) of cholesterics can be of the same order of magnitude as the wavelength of visible light. If so, a visible Bragg re¯ection occurs. On the other hand, the helix pitch is very sensitive to the in¯uence of external conditions. A combination of these properties leads to the unique optical properties of cholesterics which are of both scienti®c and practical interest. The in¯uence of electric ®elds and light irradiation on the optical properties of cholesteric structures are reviewed in this chapter. In Section 6.1 we consider the general optical properties of CLCs, such as Bragg di¨raction due to the periodical structure, refractive indices, and induced circular dichroism. In the subsequent sections, electro-optic e¨ects and light-induced e¨ects are described. Several types of electro-optic e¨ects have been observed in cholesteric liquid crystals [1]±[3]. Section 6.2 of this chapter is focused on some of the most recent results, e.g., bistability, color change e¨ects, ``amorphous cholesterics,'' and the ¯exoelectric e¨ect. In addition, dielectric, hydrodynamic, and ¯exoelectric instabilities, as well as domain structures in cholesterics, are discussed brie¯y. Light-induced molecular reorientations and optical nonlinearities in transparent (nonabsorbing) and absorbing cholesterics, photostimulated shift of the pitch, optical bistability, and optical switching are presented in Section 6.3. The generation of higher harmonics and laser generation are considered, too.

6.1

General Optical Properties of Cholesteric Liquid Crystals

The cholesteric phase appears in organic compounds which consist of elongated (nematogenic) molecules without mirror symmetry (chiral molecules) [1]±[3]. Typical representatives of these compounds are the derivatives of 159

160

G. Chilaya

cholesterol. Thus, chiral nematic liquid crystals are generally called cholesteric liquid crystals (CLCs), although the name chiral nematic is more correct. The cholesteric structure occurs not only in pure chiral compounds, but also in mixtures of achiral nematics with optically active (chiral) mesogenic or nonmesogenic dopants (induced cholesteric systems) [4]±[9]. Locally, a cholesteric is very similar to a nematic material. However, the direction of the preferable orientation of the molecules n (director) varies periodically in space. If the helical axis is oriented along z, the director n is given by (nx ˆ cos y, ny ˆ sin y, nz ˆ 0), where y ˆ q0 z ‡ constant, q0 is the wave number …q0 ˆ 2p=P†, and P is the helix pitch. The spatial period is equal to one-half of the pitch (because of the unpolarity of the cholesteric structure, see Figure 6.1). The helix may be right- or left-handed, depending on the absolute con®guration of the molecules. In some mixtures a helix sign inversion is observed when either the temperature or the concentration of the components is changed [10].

Figure 6.1. The arrangement of (a) the molecules and (b) the optical indicatrix in the cholesteric phase.

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

161

In various cholesteric systems, the period of the supermolecular structure (helical pitch) varies by a wide range (from @ 0.1 mm to several hundred mm). For the case of long pitch (low chirality) P g l (where l is the wavelength of light), the light propagating parallel to the helical axis may be described by a superposition of two eigenwaves having electric ®eld vectors parallel and perpendicular to the director. The long pitch case was studied for the ®rst time by C. Mauguin [11]. This type of con®guration can be obtained by the mechanical twist of nematics and is used in conventional twisted nematic displays [12]. In this case the structure behaves as a polarizing waveguide: the plane of polarization of linearly polarized modes follows the twist. For short pitch (high chirality), when l and P are comparable, the eigenwaves become elliptical, and in the limiting case circular. In this limiting case, selective re¯ection occurs due to Bragg di¨raction at a wavelength lB with mlB ˆ Pn cos j:

…6:1†

Here, m is the di¨raction order, j is the angle of light incidence, and n is the refractive index of the medium. The di¨raction in CLCs is responsible for some remarkable optical properties. The following characteristic features occur for light propagating along the axis of the helix. (a) Only the ®rst-order Bragg re¯ection is possible in this case. (This is con®rmed by both experimental results and theoretical considerations.) According to (6.1), the maximum of selective re¯ection occurs at the wavelength lB ˆ Pn. The spectral width of the selective re¯ection band is equal to Dl ˆ PDn, where Dn ˆ ne ÿ no is the birefringence of a nematic layer perpendicular to the helix axis. The re¯ected light is circularly polarized and the sign of rotation coincides with the sign of rotation of the cholesterics helix. (b) On each side of the selective re¯ection band there are regions with a strong rotation of the plane of polarization of light. The rotatory power amounts to more than hundreds of revolutions per mm. The rotation of the plane of polarization depends strongly on the wavelength of incident light, and an anomalous dispersion of the rotatory power is observed. According to [13] the rotation angle is given by j ˆ ‰2p d=P…ne2 ÿ no2 =ne2 ‡ no2 † 2 Š‰1=8…l 0 † 2 1=1 ÿ …l 0 † 2 Š:

…6:2†

(c) Close to the Bragg wavelength lB , the optical rotation becomes very large and changes its sign at l ˆ lB . The theory of the propagation of light along the optic axis was considered in [11], [13]±[17]. The kinematical approach could explain many experimental results. However, for quantitative explanation of the experiments, a more detailed consideration is necessary. For this purpose, the dynamical theory should be used [15]. Precise measurements of the re¯ection spectrum from a monodomain CLC show good agreement with theory [18]. A solution

162

G. Chilaya

of Maxwell's equations was ®rst given in [13]. The exact solution of Maxwell's equations for the general case is given in [16]. For the case of oblique incidence, ®rst- and higher-order di¨ractions are permitted. The polarization becomes elliptical in this case. Precise measurements of the re¯ection spectrum for the oblique incidence of this light were given in [19]. First- and second-order re¯ection spectra were calculated and observed. The analytical solution of Maxwell's equations by the dynamical theory of di¨raction [20] is in good agreement with experimental data [19]. The exact solution of Maxwell's equations has not been yet developed, because the theory is very complicated. The propagation of light perpendicular to the optical axis was studied in [21]. For a certain polarization of incident light, the cholesteric phase can be considered as a medium with a periodic gradient of the refractive index. The refractive index changes between ne and no and the period is half of the pitch. The periodicity in the phase and amplitude causes a di¨raction of polarized light. This di¨raction was used for investigating the temperature-dependence of the pitch. When the pitch of the CLC is larger than the wavelength of the visible light and if the linear birefringence is also large, it is possible to observe the forward di¨raction [22]. For a cholesteric layer between crossed polaroids, the presence of forward scattering is manifested in the form of selective dependence of the transmission coe½cients on the wavelength of light. Experimental studies on a well planar oriented CLC with certain parameters con®rm the forward di¨raction e¨ect [23].

6.1.1

Orientational Order Parameter and Refractive Indices

The cholesteric phase is thermodynamically equivalent to the nematic phase. Both phases can be characterized by an orientational order parameter S :ˆ hP2 …cos y†i, where P2 is the second Legendre polynomial, and y is the angle between the long molecular axis and the local director n [24]. The presence of twist in the cholesteric phase complicates the problem of measuring S. Many of the methods applied successfully to nematics are not suitable for CLCs. Nevertheless, some measurements using optical methods were done in order to estimate S in CLCs [25]±[29]. A nematic liquid crystal with a uniform alignment of the director n behaves like a uniaxial crystal with positive optical anisotropy ne > no (where ne 1 nkL is the refraction index for the extraordinary beam and no 1 n?L is the refraction index for the ordinary beam). We can consider the cholesteric structure as a special case of a nematic structure when the director n describes a helix. As is shown in Figure 6.1, the optical anisotropy in CLCs is negative, i.e., noh > neh , where neh 1 nkh and noh 1 n?h are the refractive indices for the extraordinary and ordinary beams, respectively. The index h indicates that the macroscopic optical axis corresponds to the direction of

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

163

the pitch axis. If the local nematic refractive indices are given by ne0 and no0 , the average refractive indices with respect to the helix axis h can be written as neh ˆ no0 , noh ˆ …ne02 ‡ no02 † 1=2 . Precise measurements of the refractive indices of racemic (nematic) and the optically active (cholesteric) form of the same liquid crystal show that the refractive indices ful®ll this theoretical expectation within experimental error (G0:0005) [30], [31]. The optical anisotropy of the liquid crystal phase is determined by the anisotropy of the polarizability of the molecules and by the degree of their order which is described by the order parameter S. The obtained results indicate that the orientational order parameter S in a layer of a CLC is essentially the same as in the nematic phase occurring in the corresponding racemic mixture.

6.1.2

Induced Circular Dichroism and Suppression of the Absorption

If a small amount of dye, possessing linear dichroic absorption, is dissolved in a CLC, the helical arrangement transforms the linear dichroism into a circular one. The induced circular dichroism is given by D ˆ …Il ÿ Ir †= …Il ‡ Ir †, where Il and Ir are the light transmission coe½cients for leftand right-handed circularly polarized light, respectively. Induced circular dichroism was studied by several authors [32]±[37]. It was investigated even for a dye possessing, simultaneously, both positive and negative dichroism [37]. With the sign inversion of the linear dichroism, the sign of circular dichroism is changed, as well. The data given in [32]±[37] were obtained for structures where the pitch was greater than the absorption wavelength of the dye. However, the absorption is suppressed when the wavelengths of absorption and selective re¯ection coincide. Since the arrangement of the molecules is helical, the absorption of light with the di¨racted polarization undergoes an abrupt change near the region of selective re¯ection. On the short wavelength side of Bragg re¯ection, the electric vector is perpendicular to the long axis of the molecules, i.e., to the absorbing e¨ective oscillators in dyes with positive dichroism. As mentioned in [2], [38], this e¨ect is analogous in many respects to the anomalous absorption of X-rays (Borrmann e¨ect) that occurs as a result of di¨raction in ordinary crystals. Some measurements and the corresponding theoretical works [38]±[41] show a quantitative agreement between the theory and the experiment [41].

6.2

Electro-optics of Cholesteric Liquid Crystals

Several types of electro-optic e¨ects have been observed in CLCs, depending on the surface treatment (boundary conditions), the helical pitch P, the thickness-to-pitch ratio d=P, the dielectric anisotropy De, and the frequency of the applied ®eld. Some of these electro-optic e¨ects are caused by texture

164

G. Chilaya

Figure 6.2. Textures in CLCs: (a) planar, (b) ®ngerprint texture, (c) focal conic, and (d) ``amorphous'' cholesteric texture.

changes. Figure 6.2 shows the the orientation of the liquid crystal for the typical cholesteric textures, namely the planar (Grandjean), focal conic, ``®ngerprint'' and ``amorphous'' cholesteric texture. In order to study electrooptic e¨ects, the liquid crystal is usually sandwiched between two parallel plates with transparent electrodes. Cholesteric liquid crystals with negative dielectric anisotropy De < 0 show a dynamic scattering for electric ®elds with low frequency [42]. A transparent ®eld-o¨ state can be obtained by preparing a planar texture. The electric ®eld induces hydrodynamic instabilities which cause a di¨use scattering appearance. Under appropriate conditions, the scattering state is maintained after removal of the ®eld, thereby providing an optical memory e¨ect (``storage mode''). The stored scattering state can be erased by the application of a high-frequency electric ®eld which reorganizes the planar structure. The hypothesis that this e¨ect is connected with the transition of a confocal texture to a planar texture was ®rst expressed in [43]. Experiments show that the erasure frequency is inversely proportional to the dielectric relaxation time [44]. In CLCs with positive dielectric anisotropy, an electric ®eld-induced cholesteric±nematic phase transition was theoretically predicted [45], [46] and experimentally observed [47], [48]. If the electric ®eld E is applied perpendicular to the helix axis h of a CLC, the helix unwinds like in a magnetic ®eld (Chapter 2). At su½ciently high ®eld strengths, the homeotropic nematic structure is stabilized (Figure 6.3). The critical ®eld strength E ˆ ECN depends on the pitch P, the dielectric anisotropy De, and the twist elastic constant K22 : ECN ˆ …p 2 =Po †…K22 =eo De† 1=2 :

…6:3†

The critical electric ®eld given by (6.3) in the analogous expression to the critical magnetic ®eld strength given by (2.22). If the electric ®eld is applied parallel to the helix, the situation is more complicated. For a short pitch, a shift of the re¯ection peak to shorter wavelengths (blue shift) can be observed [49]±[52]. It was assumed that above a threshold ®eld, a conical deformation of the planar texture leads to a contraction of the pitch, and thus lB is shifted to shorter wavelengths [52]. However, it could be shown that the blue shift of the selective re¯ection

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

165

Figure 6.3. Schematic representation of the cholesteric±nematic phase transition.

results from a periodic distortion of the texture, rather than from a pitch contraction [46]. Using the continuum theory, a ®eld-induced pitch gradient in the CLC cell has been proposed [53]. The latter e¨ect would result in a shorter pitch, which provides an alternative explanation for the ®eld-induced wavelength shift. Experiments on dual-frequency addressable mixtures with a low-crossover frequency [54], [55] lead to new conclusions on the origin of the blue shift. It was shown that both a blue shift and a red shift occur in the same system. These two processes have di¨erent relaxation times. A reversible color change can be realized by switching between the low and high frequency, especially in the more stable systems with shorter pitch. The helix pitch is very sensitive to external in¯uences and, in particular, it may strongly depend on the temperature. The property of CLCs to change the re¯ected color with temperature (due to the temperature-dependence of the pitch) has been known for a long time, and CLCs are successfully used as thermochromic material [56]. The temperature-dependence of pitch is quite complex and so far not completely understood [57]. The temperaturedependence of pitch is an advantage for thermometric applications, but it can be a major problem for the use of CLCs as an electro-optic material. In the latter case, it is necessary to obtain mixtures with an operating voltage which is independent of the temperature. For this purpose, systems with a designed pitch-temperature dependence P…T† are needed. The critical ®eld strength for the cholesteric±nematic phase transition depends on the pitch, the twist elastic constant, and the dielectric anisotropy [(6.3)]. It was shown [58], [59] that a small negative temperature coe½cient dP=dT of the pitch is suitable to compensate the temperature-dependencies of the twist elastic constant and the dielectric anisotropy, so that the temperature-dependence of ECN becomes negligible. Five di¨erent ways to obtain a temperatureindependent ®eld strength ECN in systems with an induced spiral structure are described in [60].

166

G. Chilaya

Figure 6.4. Transmittance hysteresis curve for the cholesteric±nematic phase transition.

6.2.1

Bistability and Color Change E¨ects in Cholesterics

For normal (homeotropic) surface orientation, a bistability e¨ect appears, i.e., the intensity-versus-voltage curve shows a large hysteresis [61] (Figure 6.4). As a matter of fact, the occurrence of a bistability is possible in all cases of surface treatment (including nontreated, nonrubbed cells), provided that the cell thickness d of the cholesteric sample is comparable to the helical pitch P [4], [62]. The appearance of certain textures depends on the boundary conditions, the value of the pitch, the thickness-to-pitch ratio, and the regime of the applied voltage. A deformed spiral superstructure, the so-called strain (scroll) texture, is organized under certain conditions [63], [62]. Various pitch±thickness ratios have been investigated [64], [65]. A CLC doped with dyes can even show a tristability e¨ect if the pitch is large [66]. The expression (6.3) for ECN was calculated for in®nitely thick ®lms without taking into account the boundary conditions. However, the cholesteric to nematic phase transition was investigated for di¨erent thickness [67], [68]. The in¯uence of the surface orientation was taken into account [68] by introducing a surface free energy per unit area F which leads to the following expression for VCN : VCN ˆ …8p 2 d 2 K22 =…Po2 eo De† ÿ 8Fd=…eo De†† 1=2 ;

…6:4†

where P0 is undisturbed pitch. The anchoring energy has a remarkable in¯uence on VCN [69]±[71]. The bistability behavior of the cholesteric±nematic transition was investigated for di¨erent treatments of the surface [72]. For this purpose, glass plates with transparent electrodes were coated by polyimid ®lms, and three kinds of cells were investigated with:

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

167

(1) both sides unrubbed; (2) one side rubbed; and (3) both sides rubbed. Additionally, the rubbing strength was controlled. The largest value of the hysteresis width was obtained when both sides were unrubbed. The bistability discovered in [61] was also used for practical applications. The strain texture, which possesses the e¨ect of storage was applied in image converters [73]. A color projection display has been developed based on the bistability of the cholesteric-nematic phase transition [74], [75]. The cholesteric-nematic phase transition e¨ect is also successfully used in guest±host dichroic liquid crystal displays [76]. The phenomenon of aligning the dye molecules (or other doping molecules) by the liquid crystal matrix (Section 6.1.2) is called a guest±host interaction. Consequently, liquid crystal displays (LCDs) containing oriented dye molecules are commonly known as guest±host dichroic LCDs. The application of an electric ®eld changes the absorbing state into a nonabsorbing state or vice versa. The observed contrast ratio is a¨ected by the dichroic ratio, the concentration of the dye molecules, and the cell thickness. For a given thickness, a higher dye concentration leads to a higher contrast ratio, but the corresponding transmission is reduced. For a comprehensive review on this subject, see [77]. Another application of the cholesteric±nematic phase transition was discovered recently [78]. The spectral transmission characteristics were investigated for di¨erent incident and observing angles of a CLC with a pitch P V 0:8 mm. In this case, the wavelength of selective re¯ection is not in the visible range and the initial planar texture is transparent. At the electric ®eld strength ECN , the cholesteric±nematic phase transition takes place. After decreasing the applied electric ®eld strength below ECN , but not below a critical value EFC , one observes a state with uniform color. (The critical value EFC corresponds to the transition between the focal conic texture and the Grandjean texture.) For the Grandjean texture, the angular dependence of the wavelength lB of selective re¯ection (i.e., Bragg re¯ection) is characterized by dlB =dy 0 < 0, where the angle y 0 describes the direction of observation with respect to the surface normal. However, the ®eld-induced state occurring between ECN and EFC shows an angular dependence with dlB =dy 0 > 0, which is characteristic of a di¨ractive grating. The Bragg scattering from a polydomain structure for oblique incident light in re¯ected geometry was studied previously [79], [80]. In [81], the angular dependence of the wavelength of the di¨racted light was analyzed in greater detail. The geometry of observation of the scattering characteristics is shown in Figure 6.5. In the case of a polydomain structure, the angle j occurring in the Bragg condition [(6.1)] has the meaning of an e¨ective scattering angle. If the di¨raction pattern is studied in a transmission geometry (Figure 6.5(a)), the re¯ection condition with respect to the cholesteric planes, 0 g, and jin ˆ jout , is only ful®lled if the angle j is given by j ˆ 12 fyLC ‡ yLC

168

G. Chilaya

(a)

(b)

Figure 6.5. (a) Schematic representation of the setup for investigating the scattering properties. (b) Theoretical angular dependence of lB . The values y 0 < 90 correspond to the transmission and y 0 > 90 to the re¯ection mode. Note that the two corresponding scales are very di¨erent.

thus [79]:

  lB ˆ pn sin 12farcsin…sin y=n† ‡ arcsin…sin y 0=n†g :

…6:5†

Equation (6.5) describes the angular dependence of the scattered wavelength l in the di¨raction limit, y 0 < 90 . For the case of observation in the re¯ection mode …y 0 > 90 †, the e¨ective angle j is given by j ˆ 12 fyLC ÿ 0 †g, and lB is given by [80]: …180 ÿ yLC   …6:6† lB ˆ pn cos 12farcsin…sin y=n† ÿ arcsin…sin y 0 =n†g : The calculated angular dependence is shown in Figure 6.5, as well. This ®gure describes all cases observed in the experiments very well. The color of the ®eld-induced state varies with the applied voltage in [78]. This electric ®eld-controlled color-change e¨ect was also observed in polymer-dispersed cholesteric ®lms [78] and gels [81], and is very promising for applications. Besides the voltage-controlled color e¨ect, the observed structure o¨ers another practical possibility: a laser beam passing through this structure can be de¯ected. For an He±Ne laser beam with oblique incidence (l ˆ 632 nm, j ˆ 45 ), a voltage-controlled variation of the de¯ection by about 30 was realized [78].

6.2.2

Electro-optics in Cholesterics with Medium Pitch and Amorphous Cholesteric Structure

Close to the Bragg wavelength lB , a CLC behaves ``pure'' optically active, i.e., it shows no linear birefringence. The rotation angle j of the polarization plane of normally incident light is given by (6.2). This case is denoted as a CLC with medium chirality (pitch) [82]±[86]. According to expression (6.3)

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

169

for the cholesteric±nematic transition, a low control voltage can be obtained by a large spiral pitch. However, the Mauguin region is approached on increasing P. This is evident by an increase in the optical anisotropy (linear birefringence) of the structure. To avoid uncontrolled deformations of the pitch caused by the cell walls, called strain textures [62], [63], the condition P > d must be satis®ed. Estimating the in¯uence of linear birefringence near the Mauguin region, the general conditions for the optically active structure are l V DnP=2 and P > d [84]. Most frequently, a planar orientation of CLCs is obtained by the rubbing technique. However, in cells coated with polyamide (without rubbing), a new cholesteric state, called an ``amorphous'' cholesteric structure, is formed [82]±[85]. The surface, covered by polyamide, orients the molecules parallel to the surface, but without any preferable direction in the surface plane. Thus, a structure with random orientation of liquid crystal molecules, and a helical axis oriented normal to the surfaces, is obtained (Figure 6.2(e)). All parts of the amorphous cholesteric structure with medium chirality rotate the polarization plane by the same angle. The whole structure can be considered as optically active. In this case, the angle of rotation is independent of the direction of the polarization plane of the incident light. This is the basic property used for electro-optic application of the amorphous cholesteric structure with medium chirality. Characteristic features of this e¨ect are: (1) (2) (3) (4)

low demands on the surface conditions (nonrubbed cell); an ability to function in any position between crossed polarizers; wide and uniform viewing angle; and rise times less than 10 ms and decay times of 12±20 ms.

The electro-optic e¨ect was also studied in polymer dispersed liquid crystal (PDLC) ®lms [86]. With respect to nonchiral nematic PDLC ®lms, the transmission is lower, but the angular dependence is improved.

6.2.3

The Flexoelectric Electro-optic E¨ect in CLCs

It is well known that nematic liquid crystals are nonpolar. However, for a certain asymmetrical shape of the molecules, splay or bend deformations of the director ®eld lead to an electrical polarization [87]. This feature is known as the ¯exoelectric e¨ect. Theoretically, the in¯uence of an electric ®eld on CLCs for the case where the helical axis is oriented parallel to the plane of the sample was ®rst considered by Goossens [88]. Experimentally, the ¯exoelectric electro-optic e¨ect in CLCs can be observed in conventional sandwich cells with transparent electrodes when the helix axis of the CLC lies parallel to the glass surfaces [89]. In the absence of an electric ®eld, the CLC behaves as a uniaxial material with its optic axis perpendicular to the director and parallel to the helix axis. When an electric ®eld is applied normal to the pitch axis, the helix distorts, as shown in Figure 6.6. Thus, the optical axis is reoriented and the medium becomes biaxial. The deviation direction

170

G. Chilaya Figure 6.6. Flexoelectric e¨ect: The pattern of the director rotations, induced by an electric ®eld, applied perpendicular to the plane of the drawing.

changes with the polarity of the ®eld (Figure 6.6), and the corresponding angle is approximately proportional to the electric ®eld strength. The observed ¯exoelectric e¨ect can be explained by a linear coupling between the electric polarization and splay or bend deformations of liquid crystals. The following conditions are necessary to observe this e¨ect: (1) The homogenous alignment of the pitch axis is important in one direction, i.e., the formation of a uniform lying helix (ULH) texture. (2) The pitch of CLCs should be smaller than the wavelength of the incident light, since in this case one can consider the ULH texture as a uniaxial plate. Moreover, at a larger pitch, the di¨raction of light could suppress the ¯exoelectric e¨ect. (3) To achieve large deviation angles, it is necessary to minimize the contribution of the dielectric coupling (the unwinding of the helix) and thus the dielectric anisotropy of the materials should be small. The switching time is independent of the magnitude of the applied electric ®eld and the value is about 100 ms [90]. The static and dynamic properties of the ¯exoelectric e¨ect in CLCs are described in [91], [92]. Considerable e¨orts were made in order to optimize the respective parameters of the liquid crystals [93], [94]. The ¯exoelectric e¨ect was also studied in CLCs with a temperature-induced sign change of the dielectric anisotropy. In this way it was possible to perform measurements in the regions where De G 0. The dependence of the sign of the ®eld-induced deviation of the optic axis on the handedness of the helix was established, which supports once more a ¯exoelectric origin of observed e¨ect. In CLCs showing a temperature-induced twist inversion, it was possible to separate two linear electro-optic e¨ects in the vicinity of the inversion temperature, namely the electroclinic and the ¯exoelectric e¨ect. The reorientation of the optical axis due to the electro-

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

171

clinic e¨ect is about 100 times smaller and 100 times faster than the ¯exoelectric e¨ect [95].

6.2.4

Instabilities and Domains

Several cases of dielectric, hydrodynamic, and ¯exoelectric instabilities and domain structures have been observed and extensively studied in CLCs. Their appearance depends on the initial orientation of molecules, the physical parameters of the material, and the applied electric ®eld. In CLCs with positive dielectric anisotropy De > 0, an electric ®eld applied along the helix axis of a planar (Grandjean) texture can induce a two-dimensional spatially periodic deformation which has the form of a square grid [96]. The period and threshold voltage of this ®eld-induced instability depend on the elastic constants, the dielectric anisotropy, and the sample thickness [97]. More frequently, the electrohydrodynamic instabilities were studied in CLCs with negative dielectric anisotropy, De < 0. The instability is caused by the torque induced by electrical conductivity acting against the elastic torque of the CLC. Similarly, instabilities are observed in the dielectric regime occurring at high frequencies. The threshold voltage increases with increasing frequency of the applied electric ®eld and the critical frequency is directly proportional to the conductivity [97], [98]. At higher voltages, turbulence sets in and a dynamic scattering e¨ect is observed. On switching o¨, the liquid crystal relaxes to the focal conic texture (storage mode) [42]. Theoretical and experimental results about the appearance of ¯exoelectric domains are given in [99]±[102]. The investigation of instabilities is often used for the determination of liquid crystal parameters, mostly the elastic constants. For more detailed discussions about the dielectric, hydrodynamic, and ¯exoelectric instabilities and domain structures in cholesterics, see [103], [104].

6.3

Light-Induced E¨ects in Cholesteric Liquid Crystals

Light-induced orientational nonlinearities and related e¨ects were extensively studied during the last two decades. The results have been reviewed in [105]±[108]. Several parameters of liquid crystals can change due to the in¯uence of the light ®eld. At low light intensity, the e¨ect could be due to the following processes: (1) (2) (3) (4)

change of the molecular conformation; change of the molecular interaction; local recrystalization; or photo-induced charge formation.

The nonlinear optical e¨ects, observed at high intensity of the light, are investigated for application aspects and are used in conventional nonlinear optics methods and schemes. Among them are: self-focusing, degenerate

172

G. Chilaya

four-wave mixing, optical ®eld-induced birefringence, optical bistability, and switching. Light-induced changes of the complex birefringence, i.e., changes in the refractive index or absorption for a given polarization, can be due to Kerr-like nonlinearities. Due to the helical arrangement of the director, CLCs show a helically distributed feedback and thus provide numerous possibilities for the investigation of nonlinear optical e¨ects. Photo-induced e¨ects were observed:

. . . . . .

in absorbing or nonabsorbing materials; in strong or weak ®elds; in connection with a director reorientation or without director reorientation; in setups with or without mirror; under resonance or nonresonant conditions; and due to Kerr-like or non-Kerr-like e¨ects.

Some of these e¨ects are considered in the following sections.

6.3.1

Photo-Stimulated Shift of Pitch in Absorbing Cholesteric Liquid Crystals

Among absorbing liquid crystals, most of the studied systems consist of conformationally active molecules which are capable of trans±cis (E±Z) isomerization. Typically, in this case, an elongated rod-like molecule (trans isomer) transforms under the in¯uence of ultraviolet (UV) radiation into a bent or fractured form (cis isomer). In CLCs, this change of molecules can cause a change of pitch, and thus a shift of the wavelength of selective re¯ection. Consequently, a reversible color shift was observed in conformationally active dye-doped CLCs [109]. As a dye dopant both the cis and trans azobenzene were used. A trans±cis conversion can be accomplished by irradiation with the wavelength l ˆ 313 nm and cis±trans conversion with l ˆ 420 nm. Mixtures of cholesteryl chloride (CC) and cholesteryl nonanoate (CN) were used as a cholesteric solvent. The components do not absorb above 270 nm and photochemical decomposition can be ruled out. Two mixtures were studied: in experiments with pure trans azobenzene the weight ratio CC:CN was 25:75 (dP=dT < 0, according to [110]), and with pure cis azobenzene the weight ratio CC:CN was 35:65 …dP=dT > 0†. The concentration was chosen so that the change of P due to irradiation exhibited the opposite sign of dP=dT, in order to exclude the in¯uence of heating. The dependence of lB on the concentration of pure cis and trans azobenzene was studied. A rather small pitch p was observed for trans azobenzene, and a larger pitch for cis azobenzene. Consequently, the wavelength of selective re¯ection lB is shifted to larger values if a trans azobenzene-doped mixture is irradiated with light of 313 nm, whereas lB is shifted to smaller values if a cis azobenzene-doped mixture is irradiated with light of 420 nm. Experiments show that the shift can reach 60 nm. Similar

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

173

values of the light-induced shift of lB were observed for cholesteric systems doped with another conformationally active dye [111]. A color shift can also be observed if the cholesteric compound itself is conformationally active [112], or if a photosensitive chiral compound is used as chiral dopant in an induced cholesteric mixture [113], [114], for example, a reversible red-shift of lB due to the photo-transformation of an optically active dopant was induced by an He±Cd laser beam (l ˆ 0:44 mm). The latter e¨ect can be attributed to a light-induced change of the helical twisting power [3] of the chiral dopant. The same e¨ect was observed in induced cholesteric polymers, too [115]. A photochemically induced cholesteric± nematic phase transition can occur in induced cholesteric systems [116] if the chiral dopant shows racemization under the in¯uence of UV radiation. For image formation due to UV irradiation, CLCs can be applied which show a transition to the isotropic phase [117] or an irreversible photo±chemical reaction connected with a change of pitch [118]. The high sensitivity of the pitch to light irradiation is very promising for optical data-storage applications.

6.3.2

Generation of Higher Harmonics in Cholesteric Liquid Crystals

The generation of higher harmonics is of particular interest in the ®eld of nonlinear optics. The nonlinear optical susceptibilities are due to the hyperpolarizabilities of the individual molecules and depend strongly on the molecular arrangement in the respective liquid crystalline phase. CLCs are not centrosymmetric. Thus, they can be expected to show to second harmonic generation. Experimental investigations of laser-induced second harmonic generation in CLCs are described in [119]. However, in [120] it was mentioned that this e¨ect might be connected to the presence of unmelted crystals in quasi-equilibrium with the cholesteric phase. There is no evidence of any discernible second-harmonic signal, suggesting that the molecular arrangement in these materials has an overall inversion symmetry. Third-harmonic generation in liquid crystals is clearly not forbidden by symmetry. In [121], independent con®rmation of the second-harmonic results of [120] and ®rst measurements of third-harmonic generation are reported. Large changes of the third-harmonic intensity occur at phase transitions. However, the attempt to achieve phase matching of third-harmonic generation in CLCs has not been successful in this ®rst study. The observation of third-harmonic generation was also realized in [122]. A mode-locked Nd:glass laser was used as the fundamental pump source. A typical pulse train lasted about 200 ns with the individual pulses separated by about 7 ns. The total energy in each train was about 0.03 J and the pulse width of the individual pulses was about 7 ps. In this work, it was shown that for light propagating along the helical axis of a CLC, third-harmonic generation is possible for 15 di¨erent phase-matching conditions. The following cases were considered:

174

G. Chilaya

(1) both the fundamental and the phase-matched third-harmonic waves propagate in the same direction; or (2) the phase-matched third-harmonic wave is generated in a direction opposite to the fundamental wave; or (3) the third-harmonic generation occurs when the fundamental waves are propagating simultaneously in both the forward and backward directions. For a mixture of cholesteryl chloride and cholesteryl myristate (1.75 to 1 by weight), the satisfaction of the phase-matching condition 3k…o† ˆ k…3o† was predicted [123]. This mixture shows a helix inversion. Since the pitch of the sample can vary from left- to right-handedness, there should be two phase-matching peaks. Experimental results of phase-matched thirdharmonic generation are in good agreement with the theoretical calculations. The dependence of the nonlinear conversion e½ciency on the sample thickness D, and the absolute value of the third-order nonlinear susceptibility were investigated in [124]. The temperature-dependence of the pitch was used to achieve the phase-matching conditions for the third-harmonic generation. A drop in e½ciency of the third-harmonic generation was observed for cells with D > 10 mm, because of the inhomogenity of the pitch. Nonlinear-optical frequency conversion under conditions of selective re¯ection of the generated radiation was investigated theoretically in [125]. Third-harmonic generation was considered both for light propagating along the pitch axis and oblique to the pitch axis. For oblique incident light, the polarization characteristics of the generated waves are more complicated, owing to the more complicated linear optical properties of the CLC in this case. For both cases of harmonic generation, the e½ciency of nonlinear frequency conversion can increase strongly and reach a maximum at the boundary of the selective re¯ection region. According to [125], the phasematching conditions can be ful®lled for suitable CLCs and suitable pump conditions.

6.3.3

Theoretical Aspects of Pitch Dilation, Orientational Nonlinearities, and Optical Bistability in Cholesterics

The in¯uence of circularly polarized light with high intensity on the pitch of CLCs was theoretically studied in [126]. The wavelength of the light was assumed to be far from the Bragg condition. It was found that the pitch of the CLC increases under the in¯uence of such waves. Dilation of the pitch for light with a wavelength coinciding with the Bragg wavelength was considered in [127]. In this case, the light-induced change of the pitch leads to a mirrorless bistability. The physical principle of this bistability is similar to the predicted e¨ect for distributed-feedback structures and in the degenerate four-wave mixing process for an intensity-dependent refractive index medium [128], [129]. Like a static electric ®eld, the time-averaged optical ®eld couples

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

175

to the local dielectric anisotropy and exerts torques within the CLC which may compete with the elastic torques determining its internal structure. The resulting increase of the pitch causes a change of the transmission at higher intensities. When the intensity is reduced, the appearance of the re¯ection occurs for smaller intensities than the disappearance for increasing power of light. Thus, optical bistability takes place. This bistability is a result of the light-induced pitch dilation and occurs even in the absence of external re¯ectors. The calculated critical intensity for the incident circularly polarized light is given by jEin j 2 ˆ …o=c† 2 …ea =eo e†K22 ;

…6:7†

where ea ˆ ee ÿ eo . Using typical values, ea =e ˆ 0:1, K22 ˆ 10ÿ12 N, and an incident wavelength of 1 mm, a critical value of intensity of 1 MW/cm 2 was estimated. The in¯uence of the boundary conditions on the pitch change e¨ect in a CLC cell under the in¯uence of an intense optical ®eld was theoretically studied in [130]. Three cells with di¨erent anchoring conditions were considered: (1) Strong anchoring at the input side and weak anchoring at the output side. This case coincides with the situation considered in [127]. The pitch dilation is proportional to the local ®eld intensity in this case. (2) Strong anchoring conditions at both surfaces. A pitch dilation at the input side and a pitch contraction at the output side can be observed. As a result, the slope of the phase lag of the re¯ected ®eld from the CLC is slower than in a CLC in condition (1), and the re¯ectivity is nearly constant over a wide range of input intensities. (3) Weak anchoring at the input side and strong anchoring at the output side. These conditions create pitch contraction. Light-induced orientational nonlinearities in CLCs were theoretically studied in [131]. The most strong and rapid e¨ect is the director reorientation inside the unchanged periodic structure of the cholesteric. The mechanism of nonlinearity is connected with the spatially inhomogeneous reorientation of the director. The nonlinearity leads to self-focusing, self-rotation of the axis of elliptical polarization, and self-birefringence. The numerical estimations show that a light intensity of A 10 6 W/cm 2 is necessary to cause the predicted e¨ects. A qualitatively new approach, di¨erent from previous considerations [126], [131], was proposed in [132]. A nonlocal dependence of the helical pitch on the intensity of the incident light was assumed, and the re¯ection of ®elds from the cholesteric helix was taken into account. In [133], devoted to the orientational optical nonlinearity of liquid crystals, some additional e¨ects observed in cholesterics are considered: the giant optical bistabilty, grating orientational nonlinearity, pitch change due to the direct action of light; and to the thermal e¨ects, hysteresis, and optical bistability and nonlinearities connected with absorption. Here we cite some

176

G. Chilaya

remarkable data from [133]. The giant optical nonlinearity in CLCs with at least one orienting surface should not occur. Due to heating problems in CLCs, an incident light power density of @ 10 3 W/cm 2 leads to a temperature increase of @ 2  C. An optical bistability device, utilizing the thermally induced change of the pitch, and other hysteresis e¨ects are theoretically discussed. Tabirian et al. [133] conclude that the investigation of orientational optical nonlinearities is only beginning. In this subsection, we considered some theoretical problems concerning light-induced nonlinearities in CLCs. In the next subsection we will consider experimental results observed due to this nonlinearity. Of course, some of the publications quoted in the following section include theoretical calculations, as well.

6.3.4

Generation of Tunable Radiation, Optical Bistability and Optical Switching, Pitch Dilation, and Other Photo-Optical Nonlinear E¨ects Realized in Cholesterics

The selective properties of CLCs and the ability to change the pitch make it possible to construct frequency tunable lasers. The idea of building a laser with distributed feedback using CLCs was expressed in [134], but without any calculations or estimations. The theoretical analysis for a distributedfeedback laser generation in a dye-doped CLC [135] leads to the conclusion that a pump power of A 10 5 W/cm 2 is required. A generation of light in dyedoped cholesteryl chloride was obtained for the ®rst time in the work described in [136], [137]. A three-component mixture (consisting of cholesteryl chloride, cholesteryl oleate, and cholesteryl pelargonate) with a positive temperature-dependence dP=dT > 0 was used. A benzanthrone derivative was used as a dye. Harmonic distortion of the induced cholesteric structure, detected by the distributed feedback laser, was also observed and studied [138]. Photo-induced periodic distortions of the director ®eld, due to the in¯uence of a laser beam on a CLC, were found and investigated, as well [139], [140]. A cw argon±krypton ion laser with the following wavelengths and maximum powers was used: 647 nm (A 160 mW), 515 nm (500 mW), and 488 nm (400 mW). Two types of CLC were investigated: mixtures of an absorbing nematic liquid crystal (azoxybenzene compounds), liquid crystals with cholesteryl caprate, and mixtures of a nonabsorbing nematic liquid crystal (5CB) with cholesteryl caprate. In both cases, photo-induced gratings were observed, similar to the electric ®eld-induced instabilities in CLCs [96], [97]. The appearance of the square-shaped periodic distortion of the director ®eld was found to be a power threshold phenomenon. The investigations indicate that the distortions are due to conformational transformations of the molecules in the case of the absorbing CLCs, and due to inhomogeneous heating in the case the nonabsorbing CLCs.

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

177

In [141], a simple probe-pump laser technique was proposed to investigate the change of lB in CLCs under the in¯uence of intense laser radiation. An argon laser beam (l ˆ 514 nm), incident normal to the liquid crystal cell, was used as a pump beam, and an oblique incident laser beam from a laser diode (l ˆ 670 nm) as a probe beam. A nonabsorbing isothiocyanate mixture with lB ˆ 510 nm was investigated. An increase of the pitch at very low laser intensities (100±700 mW) was observed which depends on the polarization of the beam. An energy coupling between counterpropagating laser beams in CLCs, due to orientational nonlinearities, was theoretically considered and experimentally realized [142]. This interaction occurred in a case when the wavelength of the laser was close to the selective re¯ection wavelength of the CLCs. The pitch of an induced cholesteric mixture (consisting of the nematic liquid crystal 5CB and cholesteryl pelargonate) was varied by changing the concentration of the chiral dopant. With the irradiation of a ruby laser (800 ms duration) at intensities of 10±100 kW/cm 2 , an orientational backward stimulated scattering was observed. As predicted in [132], self-focusing and nonlinear optical activity of a CLC occurs only in the case of elliptic polarization. But the characteristics of the observed nonlinearities indicate a nonorientational mechanism of the observed e¨ects. A thermal mechanism is responsible for the nonlinearity in this case. It was concluded that the interaction of concurrent waves, and also self-focusing and nonlinear optical activity due to the orientational mechanism of the nonlinearity of CLCs, require high radiation intensities which exceed the damage threshold of the samples. Attempts to observe laser-induced nonlinear re¯ection in nonabsorbing CLCs were also made [143]. Cholesteric mixtures consisting of 2-methylbutyl-p-[( p-methoxy-benzilidine)amino] cinnamate and cholesteric oleyl carbonate with lB ˆ 532 nm, and mixtures of cholesteryl chloride and cholesteryl nonanoate with lB ˆ 591 nm were used. A pulsed, high-power Nd:YAG laser was applied. The series of experiments at power densities up to 4 MW/cm 2 (532 nm) and 8 MW/cm 2 (1064 nm) did not show any optically induced pitch dilation or distortion of the CLC cell. It was concluded that short laser pulses (15 ns) are not suitable to observe the light-®eldinduced pitch dilation predicted in [127]. Laser-induced gratings have been produced by mixing two laser beams at a small angle in dye-doped CLCs [144]. The sample (a mixture of n-( pmethoxy benzilidene)-p 0 -butylaniline and cholesteryl oleyl carbonate with lB ˆ 440 nm) was irradiated by two coherent argon ion laser beams. The beams were polarized in the direction parallel and perpendicular to the helix axis. A transient grating was observed at intensities of 50 mW per beam. The decay time of the transient grating was of the order of 1 second. At higher laser power, a ``persistent'' grating with a lifetime of several hours was formed. Unfortunately, the mechanism of formation and characteristic features of the gratings were not explained.

178

G. Chilaya

The dynamics of the change of the pitch in dye-doped CLCs under the in¯uence of laser radiation was considered in [145]. Three di¨erent compositions of mixtures of cholesteryl chloride, cholesteryl pelargonate, and cholesteryl oleyl carbonate with lB ˆ 650, 570, and 486 nm were used (the third mixture also contained the nematic compound 4 0 -hexyl-4-cyanobiphenyl). The mixtures have di¨erent signs of dP=dT. A copper complex with an absorbing wavelength of 355 nm was used as a dye, and the third harmonic of an Nd:YAG laser (355 nm) with 11 ns pulses was used as a light source. The shift of the selective re¯ection wavelength was measured by means of a kinetic absorption spectrophotometer. An analyzing beam was incident normal to the cell and the laser pump beam was incident at an angle of A 20 with respect to the probe beam. Three di¨erent optical responses are observed. The fastest (U200 ns) process is an increase of the pitch …lB †. This nanosecond time phenomenon was explained by the formation of a transient grating caused by the modulation of the refractive index associated with the heat instantaneously (U10 ps) released by the dye molecules to their immediate neighborhood. The second process (30±600 ms) is a change of pitch according to its temperature dependence, i.e., a thermal process. The last (>1000 ms) process is the return of the pitch to its initial value after the heat is dissipated. Optically induced detuning of the selective re¯ectivity band in absorbing CLCs has been demonstrated for applications like mirrorless optical bistability and optical switching. Optical bistability in a dye-doped CLC with a distributed feedback was for the ®rst time observed by two groups independently and simultaneously [146], [147]. An important condition for achieving optical hysteresis is the presence of a feedback. In a periodic cholesteric structure, the feedback is due to the Bragg di¨raction of light. An optical hysteresis has been achieved in dye-doped CLCs at comparatively low-light intensities [146]. A pulsed nitrogen laser with a wavelength of 337.1 nm was used in [146] and optical hysteresis has been achieved in dye-doped CLCs at light intensities as low as A 10 kW/cm 2 with pulse lengths of G 8 ns. In [147], a circularly polarized He±Ne laser (633 nm) was used as a pump and a dyedoped mixture of the nematic liquid crystal 5CB with cholesteryl pelargonate and cholesteryl oleate was investigated. The hysteresis was observed at low intensities, A 10 W/cm 2 . Transient grating optical nonlinearities in dye-doped CLCs are investigated with laser-induced dynamic grating experiments using nanosecond and picosecond laser excitation pulses [148]. The chiral nematic liquid crystal CB15 was used, which was doped by 4% of a mixture of cetocyanide dyes. The Bragg wavelength of the CLC was lB ˆ 570 nm and the absorption band of the dye was at l ˆ 550 nm. In self-di¨raction experiments, laserinduced dynamic gratings have been excited by the second harmonic of Nd:YAG laser pulses at lexc ˆ 532 nm which were obtained in an asymmetric beam-splitting arrangement. Strong self-di¨raction has been observed for laser pulses of 15 ns duration and intensities of 1.1 GW/cm 2 . A further

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

179

increase in the laser intensity results in permanent gratings. The CLC is transformed from a planar texture to the confocal texture. The latter state is stable so that an optical storage e¨ect occurs. The dynamic response of dye-doped CLCs with an absorption maximum near the selective re¯ection wavelength shows fast subnanosecond switching times. The dynamics of the grating formation was studied by using green excitation pulses of 80 ps duration, and the transient grating was probed with a continuous wave (cw) argon ion laser at a wavelength of lp ˆ 488 nm. Di¨erent response times were observed. A fast contribution to the dynamic gratings has been explained by resonant optical nonlinearity due to absorption saturation of the dye. A secondary process connected with thermal gratings has been observed as a result of radiationless recombination of the excited dye molecules. It was shown that the thermal grating gives rise to density modulations and elastic deformations of the liquid crystal host which change the e¨ective refractive index of the CLC and lead to the observed di¨raction e¨ects. The slower dynamics of the grating is explained by the relaxation of thermal gratings, density, and elastic deformations, which exhibit characteristic time constants from A 100 ns to hundreds of ms. According to a review on Kerr-like optical nonlinearities in liquid crystal [149], these experiments show that the photonic response may be much faster if short intense laser pulses are used instead of low-power cw laser radiation. This may open new aspects and perspectives in research and applications. In [150], the necessary conditions for the observation of resonance nonlinearities in dye-doped CLCs are summarized: (i) The medium should have a periodic helical structure and be thick enough to show a selective Bragg re¯ection that provides a distributed feedback. (ii) The dopant molecules should absorb light resonantly in order to provide a nonlinear response of the matter to the incident radiation. A threshold light intensity causes a change to the refractive index, which is su½cient to produce a certain phase delay and frustrate the Bragg condition. (iii) The resonance frequency of the dopant should be within the selective re¯ection band of the matrix. This enables both nonlinear response and feedback in the medium. Mirrorless optical nonlinear e¨ects (optical bistability, optical switching, and transient grating nonlinearity) as mentioned above, were observed only in dye-doped CLCs. In nonabsorbing CLCs, the realization of the lightinduced pitch dilation became possible when it was used in optical schemes as a mirror. In [151] it was shown theoretically that under intense optical radiation with a Gaussian pro®le, a retro-self-focusing e¨ect and a pinholing e¨ect occur in CLCs. These e¨ects have been observed in an experiment using a CLC dielectric resonator. The CLC dielectric resonator consists of an HR

180

G. Chilaya

dielectric end mirror, a cw-operated Nd:YAG laser, a quarter-wave plate and a CLC end mirror. When the CLC element is used as a laser end mirror, it acts like a well-aligned concave mirror±pinhole combination. The retroself-focusing e¨ects are a result of pitch dilation due to nonlinear coupling of the intense optical ®eld with the CLC structure. More detailed experimental investigation of the unique properties of a CLC as a laser end mirror were done in [152]. It was shown that a CLC mirror can be used as a laser end mirror in a solid state laser oscillator to provide both TEM00 -mode operation for cw powers in excess of 1 W. No pinhole is required to obtain TEM00 -mode operation. CLCs have been investigated as a polarizing end mirror in an optical resonator for high-power pulsed solid state lasers [153]. In a resonator for an electro-optic Q-switched Nd:YAG laser (l ˆ 1064 nm), a CLC cell with a mixture nematic liquid crystal ZLI-2359 and the chiral compound CB15 was used instead of a dielectric mirror and a polarizer. In combination with an active Q-switched laser, pulses of 10 ns duration with peak intensities up to 1 GW/cm 2 have been realized. Compared with results obtained for a conventional resonator scheme, the pulses from the laser with a CLC mirror show a somewhat weaker modulation and an asymmetry, which can be explained by a decrease in re¯ection for strong optical ®elds during the pulse. Additional measurements show that the re¯ectivity of the CLC mirrors for circularly polarized light decreases from 90% at low intensities down to 25% for intensities of A 100 MW/cm 2 . A change of the re¯ectivity occurs in connection with the unwinding of the helix in high intense optical ®elds, as predicted in [127]. The pinholing e¨ect was also experimentally observed. A change in the re¯ectivity of nonabsorbing CLC mirrors under the in¯uence of pulsed laser irradiation was also observed in [154]. A mixture of the nematic liquid crystal E7 with the chiral compound CB15 and with lB ˆ 1064 nm was used. Cells were prepared with strong anchoring on one side and weak anchoring on the other side. CLC samples irradiated by a Nd:YAG laser operated in two regimes: (1) in a cw mode; and (2) in an acousto-optically Q-switched mode, emitting 500 ns pulses with a frequency of 4.5 kHz and intensities of 10 6 ±10 7 W/cm 2 . The re¯ectivity change was observed only under pulse irradiation and only under re¯ection of the incident light from the strong-anchoring side of the cell. The CLC mirror transmittance changed between 5% and 30±80%. The observed e¨ects were interpreted within the frame of the pitch-dilation model, predicted in [127]. The results presented in this subsection show that almost all predicted nonlinear e¨ects were observed and studied in CLCs. The results are very promising for applications in information techniques and photonics. However, many physical mechanisms, especially concerning processes at the molecular level, are still to be solved. Another challenge is the light-induced pitch unwinding in nonabsorbing mirrorless CLC structures which is not clari®ed, yet.

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

181

In conclusion, CLCs are an important electro-optical and opto-optical material owing to the possibility of changing their unique optical properties easily in external ®elds. Due to their versatile behavior, many further interesting results can be expected in the future.

References [1] P.G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993. [2] S. Chandrasekhar, Liquid Crystals, Cambridge University Press, Cambridge, 1992. [3] V.A. Beliakov, Optics of Complex Structured Periodic Media, Springer-Verlag, New York, 1992. [4] G. Chilaya, Physical properties and applications of liquid crystals with induced helicoidal structure, Metsniereba, Tbilisi, 1985 (in Russian). [5] G. Chilaya, Rev. Phys. Appl. 16, 193 (1981). [6] G. Solladie and R.G. Zimmermann, Angew. Chem. 96, 335 (1985). [7] G. Gottarelli, M. Hibert, B. Samori, G. Solladie, G.P. Spada, R. Zimmermann, J. Am. Chem. Soc. 105, 7318 (1983). [8] G.S. Chilaya and L.N. Lisetski, Sov. Phys. Uspekhi 24, 496 (1981). [9] G.S. Chilaya and L.N. Lisetski, Mol. Cryst. Liq. Cryst. 140, 243 (1986). [10] G. Heppke, D. LoÈtzch, and F. Oestrecher, Z. Naturforsch. 42a, 279 (1986). [11] C. Mauguin, Bull. Soc. France Miner. Crystallogr. 34, 71 (1911). [12] M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127 (1971). [13] H. de Vries, Acta Cryst. 4, 219 (1951). [14] C.W. Oseen, Trans. Faraday Soc. 29, 883 (1933). [15] S. Chandrasekhar and K.N. Srinisava Rao, Acta Cryst. A24, 445 (1968). [16] E.I. Kats, Zh. Eksper. Teoret. Fiz. 59, 1854 (1970). [17] R. Nityanada, Mol. Cryst. Liq. Cryst. 21, 315 (1973). [18] R. Dreher, G. Meyer, and A. Saupe, Mol. Cryst. Liq. Cryst. 13, 17 (1971). [19] D.W. Berreman and T.C. Sche¨er, Phys. Rev. Lett. 25, 577 (1970). [20] V.A. Beliakov and V.E. Dmitrienko, Fiz. Tverd. Tela 15, 2724 (1973). [21] E. Sackman, S. Meiboom, L. Snyder, A.E. Meixner, and R.E. Deitz, J. Am. Chem. Soc. 90, 3567 (1968). [22] S.M. Osadchy, Sov. Phys. Crystallogr. 29, 573 (1984). [23] D.G. Khoshtaria, S.M. Osadchy, and G.S. Chilaya, Sov. Phys. Crystallogr. 30, 775 (1985). [24] W. Maier and A.Z. Saupe, Naturwissensch. A13, 564 (1958). [25] E.H. Korte, Mol. Cryst. Liq. Cryst. 44, 151 (1978). [26] D.W. Berreman and T. Sche¨er, Phys. Rev. A5, 1397 (1972). [27] A.V. Tolmachev, V.G. Tischenko, and L.N. Lisetski, Fiz. Tverd. Tela 19, 1886 (1977). [28] P. Adamski, L.A. Dylik-Gromiec, and M. Wojcechowski, Mol. Cryst. Liq. Cryst. 75, 33 (1981). [29] E.M. Averianov and V. Shabanov, Kristallogra®a 24, 184 (1979). [30] G.S. Chilaya, S.N. Aronishidze, Z.M. Elashvili, M.N. Kushnirenko, and M.I. Brodzeli, Soobshch. Akad. Nauk GSSR 84, 81 (1976).

182 [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65]

G. Chilaya G. Pelzl, Z. Chem. 17, 264 (1976). F.D. Saeva and J.J. Wysocki, J. Am. Chem. Soc. 94, 135 (1971). I. Chabay, Chem. Phys. Lett. 17, 283 (1972). E. Sackman and J. Voss, Chem. Phys. Lett. 14, 528 (1972). G. Holzwarth and N.A.W. Holzwarth, J. Opt. Soc. Am. 63, 324 (1973). H.J. Krabbe, H. Heggemeier, B. Schrader, and E.H. Korte, Angew. Chem. 89, 831 (1977). D.G. Khoshtaria, G.S. Chilaya, A.V. Ivashchenko, and V.G. Rumiantsev, 5th International Liquid Crystal Conference of Socialist Countries, Odessa (USSR), 1983, Abstracts, V1, Part 2, p. 94. V.A. Beliakov, V.E. Dmitrienko, and V.P. Orlov, Sov. Phys. Uspekhi 22, 63 (1979). V.A. Beliakov and V.E. Dmitrienko, Sov. Phys. Solid State 18, 1681 (1976). K.A. Suresh, Mol. Cryst. Liq. Cryst. 35, 267 (1976). S.N. Aronishidze, V.E. Dmitrienko, D.G. Khoshtaria, and G.S. Chilaya, JETP Lett. 32, 17 (1980). G. Heilmeyer and J. Goldmacher, Proc. IEEE 57, 34 (1969). W. Haas, J. Adams, and G. Dir, Chem. Phys. Lett. 14, 95 (1972). G.S. Chilaya, V.T. Lazareva, and L.M. Blinov, Kristallogra®ya 18, 203 (1973). P.G. de Gennes, Solid State Commun. 6, 163 (1968). R.B. Meyer, Appl. Phys. Lett. 12, 281 (1968). J.J. Wysocki, J. Adams, and W. Haas, Phys. Rev. Lett. 20, 1024 (1968). F.J. Kahn, Phys. Rev. Lett. 24, 209 (1970). W. Harper, Liquid Crystals, Gordon and Breach, New York, 1967. N. Oron and M.M. Labes, Appl. Phys. Lett. 21, 243 (1972). H. Baessler, T.M. Laronge, and M.M. Labes J. Chem. Phys. 51, 3213 (1969). C.J. Gerritsma and P. van Zanten, Mol. Cryst. Liq. Cryst. 15, 257 (1971). I. Fedak, R.D. Pringle, and G.H. Curtis, Mol. Cryst. Liq. Cryst. 64, 69 (1980). G. Chilaya, G. Hauck, H.D. Koswig, G. Petriashvili, and D. Sikharulidze, Cryst. Res. Tech. 32, 25 (1997). G. Chilaya, G. Hauck, H.D. Koswig, and D. Sikharulidze, Proc. SPIE 3318, 351 (1998). I. Sage, Thermochromic liquid crystal devices, in: Liquid CrystalsÐApplications and Uses, vol. 3, p. 302 (edited by B. Bahadur), World Scienti®c, Singapore, 1990. G. Chilaya, F. Oestreicher, and G. Scherowsky, Mol. Mat. 9, 291 (1998). A. Gobl-Wunsch, G. Heppke, and F. Oestreicher, J. Phys., Paris 40, 773 (1979). P. Gerber, Phys. Lett. 78, 285 (1980). G. Chilaya, Z. Elashvili, D. Sikharulidze, S. Tavzarashvili, K. Tevdorashvili, and K. Vinokur, Mol. Cryst. Liq. Cryst. 209, 93 (1991). W. Greubel, Appl. Phys. Lett. 25, 5 (1974). G.S. Chilaya, S.N. Aronishidze, K.D. Vinokur, S.P. Ivchenko, and M.I. Brodzeli, Acta Phys. Polon. A54, 655 (1978). M.I. Press and A.S. Arrott, J. Physique 37, 384 (1976). V.G. Chigrinov, V.V. Beliaev, S.V. Beliaev, and M.F. Grebenkin, Sov. Phys. JETP 50, 994 (1979). G. Hauck and H.D. Koswig, Cholesteric-nematic phase change, in: Selected Topics in Liquid Crystals Research (edited by H.D. Koswig), Akademie-Verlag, Berlin, pp. 115±139, 1990.

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101]

183

C.G. Lin-Hendel, Appl. Phys. Lett. 38, 615 (1981). P.R. Gerber, Z. Naturforschung 36A, 718 (1981). H.A. Van Sprang, J. L. M. van de Venne, J. Appl. Phys. 57, 175 (1985). J.C. Lee, D.W. Allender, and V.D. Ne¨, Mol. Cryst. Liq. Cryst. 210, 11 (1992). J.C. Lee and D.W. Allender, Jpn. J. Appl. Phys. 33, 1407 (1994). A. Mochizuki and S. Kobayashi, Mol. Cryst. Liq. Cryst. 225, 89 (1993). Y. Yabe and D.-S. Seo, Liq. Cryst. 17, 847 (1994). G.S. Chilaya, D.G. Sikharulidze, and M.I. Brodzeli, J. Physique C3-40, C3274 (1979). Y. Koike, A. Mochizuki, Yoshikawa, Displays p. 93 April, (1989). Y. Yabe, H. Saito, T. Yoshihara, S. Kasahara, A. Mochizuki, and T. Narusava, J. SID 1, 411 (1993). D.L. White and G.N. Taylor, J. Appl. Phys. 45, 4718 (1974). B. Bahadur, Dichroic liquid crystal displays, in: Liquid CrystalsÐApplications and Uses, vol. 3, p. 68 (edited by B. Bahadur), World Scienti®c, Singapore, 1990. G. Chilaya, G. Hauck, H.D. Koswig, and D. Sikharulidze, J. Appl. Phys. 80, 1907 (1996). F.J. Kahn, Phys. Rev. Lett. 24, 209 (1970). J.L. Fergason, Mol. Cryst. 1, 293 (1966). H.-S. Kitzerow, Jpn. J. Appl. Phys. 36, L349 (1997). K.D. Vinokur, D.G. Sikharulidze, G.S. Chilaya, and Z.M. Elashvilui, Liquid crystals with helical structure and their applications in displays, Metsniereba, Tbilisi, 1988 (in Russian). A. Chanishvili, G. Chilaya, and D. Sikharulidze, Mol. Cryst. Liq. Cryst. 207, 53 (1991). A. Chanishvili, G. Chilaya, and D. Sikharulidze, Appl. Optics 33, 348 (1994). A. Chanishvili, D. Sikharulidze, G. Chilaya, and G. Petriashvili, Mol. Mat. 8, 295 (1997). U. Behrens, H.-S. Kitzerow, and G. Chilaya, Liq. Cryst. 17, 597 (1994). R. Meyer, Phys. Rev. Lett. 22, 918 (1969). W.J.A. Goossens, J. Phys (Paris), 43, 1469 (1982). J.S. Patel and R. Meyer, Phys. Rev. Lett. 58, 1538 (1987). J.S. Patel and S.-D. Lee, J. Appl. Phys. 66, 1879 (1989). J.S. Patel, S.-D. Lee, and R. Meyer, J. Appl. Phys. 67, 1293 (1990). J.S. Patel, S.-D. Lee, and R. Meyer, Mol. Cryst. Liq. Cryst. 209, 79 (1991). L. Komitov, S.T. Lagerwall, B. Stebler, and A. Strigazzi, J. Appl. Phys. 76, 3762 (1994). P. Rudquist, M. Bujivydas, L. Komitov, and S.T. Lagerwall, J. Appl. Phys. 76, 7778 (1994). I. Dierking, P. Rudquist, L. Komitov, S.T. Lagerwall, and B. Stebler, Mol. Cryst. Liq. Cryst. 304, 389 (1997). C.J. Gerritsma and P. van Zanten, Phys. Lett. 37A, 47 (1971). J.P. Hurault, J. Chem. Phys. 59, 2068 (1973). W. Helfrich, Appl. Phys. Lett. 17, 531 (1970). B.A. Umanski, V.G. Chigrinov, L.M. Blinov, and Yu. B. Pod'yachev, Zh. Eksper. Teoret. Fiz. 81, 1307 (1981). V.G. Chigrinov, Kristallogra®ya 28, 825 (1983). E.M. Terentiev and S.A. Pikin, Kristallogra®ya 30, 227 (1985).

184

G. Chilaya

[102] L. Bourdon, M. Sommeria, and M. Kleman, J. Phys. (Paris) 43, 77 (1982). [103] L.M. Blinov and V.G. Chigrinov, Electrooptic E¨ects in Liquid Crystal Materials, Springer-Verlag, New York, 1994. [104] S.A. Pikin, Structural Transformations in Liquid Crystals, Gordon and Breach, New York, 1990. [105] I.C. Khoo and S.T. Wu, Optics and Nonlinear Optics in Liquid Crystals, World Scienti®c, Singapore, 1993. [106] P. Pal¨y-Muhoray, The nonlinear optical response of liquid crystals, in: Liquid CrystalsÐApplications and Uses, vol. 1, p. 493 (edited by B. Bahadur), World Scienti®c, Singapore, 1990. [107] B.Ya. Zeldovich and N.V. Tabirian, Sov. Phys. Uspekhi 28, 1059 (1985). [108] L.M. Blinov, J. Nonlinear Optical Phys. Mat. 5, 165 (1996). [109] E. Sackman, J. Am. Chem. Soc. 93, 7088 (1971). [110] J. Adams, W. Haas, and W. Wysocki, Phys. Rev. Lett. 22, 92 (1969). [111] B. Schnuriger and J. Bourdon, J. Chim. Physique 73, 796 (1976). [112] N.N. Goldberg and J.F. Fergason, US Patent 3,720,658 (March 1973). [113] V. Vinogradov, A. khizhniak, L. Kutulya, Yu. Reznikov, and V. Reshetnyak, Mol. Cryst. Liq. Cryst. 192, 273 (1990). [114] S.N. Yarmolenko, L.A. Kutulya, V.V. Vaschenko, and L.V. Chepeleva, Liq. Cryst. 16, 877 (1994). [115] N.I. Boiko, L.A. Kutulya, Yu.A. Reznikov, T.A. Sergan, and V.P. Shibaev, Mol. Cryst. Liq. Cryst. 251, 311 (1994). [116] C. Mioskowski, J. Bouruignon, and S. Candau, Chem. Phys. Lett. 38, 456 (1976). [117] W.E. Haas, K.F. Nelson, J.E. Adams, and G.A. Dir, J. Electrochem. Soc. 121, 1667 (1974). [118] W. Haas, J. Adams, and J. Wysocki, Mol. Cryst. Liq. Cryst. 7, 371 (1969). [119] I. Freund and P.M. Retzepis, Phys. Rev. Lett. 18, 393 (1967). [120] G. Durand and C.H. Lee, Compt. Rend. 264, Series B, 1393 (1967). [121] L.S. Goldberg and J.M. Schnur, Appl. Phys. Lett. 14, 306 (1969). [122] J.W. Shelton and Y.R. Shen, Phys. Rev. A5, 1867 (1972). [123] J.L. Fergason, Appl. Optics 7, 1729 (1968). [124] S.A. Akopian, S.M. Arakelian, R.V. Kochikyan, S.Ts. Nersesisyan, and Yu.S. Chilingaryan, Kvantovaya Elektronika 4, 1441 (1977). [125] V.A. Belyakov and N.P. Shipov, Sov. Phys. JETP 55, 674 (1982). [126] S.G. Dmitriev, Zh. Eksper. Teoret. Fiz. 65, 2466 (1973). [127] H.G. Winful, Phys. Rev. Lett. 49, 1179 (1982). [128] H.G. Winful, J.G. Marburger, and E. Garmire, Appl. Phys. Lett. 35, 379 (1979). [129] H.G. Winful and J.G. Marburger, Appl. Phys. Lett. 36, 613 (1980). [130] J.-C. Lee, A. Schmid, and S.D. Jacobs, Mol. Cryst. Liq. Cryst. 166, 253 (1989). [131] N.V. Tabirian and B. Ya. Zel`dovich, Mol. Cryst. Liq. Cryst. 69, 19 (1981). [132] B.Ya. Zel`dovich and N.V. Tabirian, Sov. Phys. JETP 82, 99 (1982). [133] N.V. Tabirian, A.V. Sukhov, and B.Ya. Zel`dovich, Mol. Cryst. Liq. Cryst. 136, 1 (1986). [134] L.S. Goldberg and J.M. Schnur, US Patent 3,771,065 (1973). [135] N.V. Kukhtarev, Sov. Quantum Electron. 8, 774 (1978). [136] P.I. Il'chishin, E.A. Tikhonov, V.G. Tishchenko, and M.T. Shpak, Sov. Quantum Electron. 8, 1487 (1978).

6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics

185

[137] P.I. Il'chishin, E.A. Tikhonov, V.G. Tishchenko, and M.T. Shpak, Sov. JETP 32, 27 (1980). [138] P.I. Il'chishin, E.A. Tikhonov, V.G. Tishchenko, and M.T. Shpak, Mol. Cryst. Liq. Cryst. 191, 351 (1990). [139] A.S. Zolot'ko, V.F. Kitaeva, N.N. Sobolev, V.Yu. Fedorovich, and N.M. Shtikov, Sov. JETP 43, 477 (1986). [140] V.F. Kitaeva and A.S. Zolot'ko, Mol. Cryst. Liq. Cryst. Sci. Technol.-Sec. B: Nonlinear Optics 2, 261 (1992). [141] M. Sierakowski, M. Karpierz, T.-L. Do, and M. Roszko, Mol. Cryst. Liq. Cryst. 282, 139 (1989). [142] T.V. Galstian, A.V. Sukhov, and R.V. Timashev, Sov. Phys. JETP 68, 1001 (1989). [143] H. Espinet and M. Lesiecki, J. Appl. Phys. 59, 1386 (1986). [144] H. Espinet and M. Lesiecki, Appl. Phys. Lett. 50, 1924 (1987). [145] R.S. Becker, S. Chakravoli, and S. Das, J. Chem. Phys. 90, 2802 (1989). [146] L.I. Zagainova, G.V. Klimusheva, I.P. Kryzhanoskij, and N.V. Kukhtarev, JETP Lett. 42, 436 (1985). [147] R.B. Alaverdian, S.M. Arakelian, and Yu.S. Chilingarian, JETP Lett. 42, 451 (1985). [148] H.J. Eichler, D. Grebe, R. Macdonald, A.G. Iljin, G.V. Klimusheva, and L.I. Zagainova, Optical Mater. 2, 201 (1993). [149] R. Macdonald and H.J. Eichler, Appl. Phys. B60, 543 (1995). [150] A.G. Iljin, G.V. Klimusheva, and L.I. Zagainova, Mol. Mat. 5, 25 (1995). [151] J.-C. Lee, S.D. Jacobs, and A. Schmid, Mol. Cryst. Liq. Cryst. 150b, 617 (1987). [152] J.-C. Lee, S.D. Jacobs. T. Gunderman, A. Schmid, and M.D. Skeldon, Optics Lett. 15, 959 (1990). [153] D. Grebe, R. Macdonald, and H.J. Eichler, Mol. Cryst. Liq. Cryst. 282, 309 (1996). [154] S.G. Lukishova, S.V. Belyaev, K.S. Lebedev, E.A. Magularia, A.W. Schmid, and N.V. Malimonenko, Mol. Cryst. Liq. Cryst. 303, 79 (1997).

7

Blue Phases Peter P. Crooker

7.1

Introduction

Although liquid crystals have been known for more than 100 years, discovering the structures of their many thermodynamic phases is an activity that persists to this day [1]. The sheer variety has been impressive, including such novelties as uniaxial, biaxial, and ferroelectric ¯uids; phases with hexatic order; and chiral phases such as the blue phase (BP) and twist grain boundary (TGB) phase, which are stabilized by a lattice of defects. Many of these phases are unique in condensed matter physics; their presence never fails to challenge our notions of how matter can arrange itself in the aggregate. This chapter reviews the present understanding of blue phases. Blue phases are distinct thermodynamic phases that appear over a narrow temperature range at the helical±isotropic boundary of highly chiral liquid crystals. In the absence of electric ®elds, there can be three blue phases: BPI and BPII, both of which have cubic symmetry; and BPIII, which possesses the same symmetry as the isotropic phase. Figure 7.1 shows schematically the phases in both nonchiral and chiral nematics. For nonchiral nematics, including racemic mixtures (with equal numbers of left- and right-handed versions of the same molecule) and even weakly chiral nematics, the nematic (or weakly chiral) phase heats directly to the isotropic phase. When the chirality is high, however, as many as three blue phases may appear. An explanation of the nomenclature should be made here. First, chiral nematic molecules need not come from cholesteryl derivatives, so we use the term chiral nematic instead of cholesteric when referring to liquid crystal materials. The chiral nematic/cholesteric phase itself we will call helical. Second, blue phases got their name from their blue appearance in early investigations. Blue phases are not always blue, however; we now know that they may re¯ect light of other colors, including near infrared. Finally, BPIII was known as the fog phase or the gray fog phase in early publications. Although these terms are descriptive of this phase's appearance, BPIII seems to have survived. 186

7. Blue Phases

187

Figure 7.1. Schematic picture of the temperature region near the nematic (N)isotropic (ISO) phase transition. Top: Nonchiral molecules have only nematic and isotropic phases. Bottom: Chiral molecules have helical (H) and isotropic phases, and, depending on the chirality, up to three blue phases (BPI, BPII, and BPIII). BPI and BPII are cubic; BPIII has the same symmetry as the ISO phase.

7.1.1

Chirality

That a liquid can have a cubic structure is truly remarkable. For a long time it was thought that a chiral nematic was just a nematic with twist, and that nothing fundamentally new was involved. As it turns out, this assumption was wrong. A nematic has orientational order, including mirror symmetry, but no positional orderÐit is invariant under a translation in any direction. When the nematic becomes chiral, the mirror symmetry is lost and the translational symmetry is reduced. Thus, in addition to becoming chiral, the chiral nematic also becomes spatially periodic. The removal of mirror symmetry permits a chiral term, previously disallowed for nematics, to now be included in the Landau free energy [2], [3], [4]. The presence of this term greatly complicates minimization of the free energy in three dimensions and ultimately leads to the blue phases, which are periodic. (Interestingly, for a four-dimensional liquid crystal the minimization can be achieved [5], [6].) Chirality, therefore, leads to new phases and new physics. The e¨ect of adding periodicity to the nematic has been described by Brazovskii [7], [8]. The nematic±isotropic transition is an example of a transition between two spatially uniform phases, which means that the transition takes place at the origin of wave-vector space. Fluctuations of the system

188

P.P. Crooker

away from the origin, which would add spatially periodic components to the structure, are energetically unfavorable and do not play a signi®cant role in the transition. However, the isotropic±helical (or isotropic±blue phase) transition is an example of a transition between a uniform upper phase and a spatially periodic lower phase with spatial period P. Such a transition takes place on a spherical shell in k-space of radius q0 ˆ 2p=P. Fluctuations away from this shell are, as before, energetically unfavorable, but ¯uctuations along the surface of the shell, which merely alter the direction of the periodic axis, but do not change the magnitude of the free energy, are now allowed, with the result is that these ¯uctuations can signi®cantly a¨ect the nature of the transition. The chiral term therefore makes the helical/blue phase± isotropic transition fundamentally di¨erent from the nematic±isotropic transition, essentially by changing the topology of the minimal free energy surface in k-space. Other transitions a¨ected by the Brazovskii mechanism are the smectic C±nematic transition and the Rayleigh±Benard instability in ¯uids.

7.1.2

Frustration

Another important feature of the blue phases is that of frustration [9], [10]. In frustrated systems, the conditions which produce a local energetic minimum cannot be extended globally. The blue phases are such systems; the lowest energy director con®guration, as we will show later, is a cylindrical doubletwist tube in which the director rotates spatially about any radius of a cylinder. Fitting these double-twist cylinders into a three-dimensional structure so that the directors match everywhere is topologically impossible, however, and so disclinationsÐline defects which appear where the cylinder directors cannot be matchedÐare necessary to relieve the elastic strain energy [11]. Figure 7.2 shows the situation for double-twist tubes with right-handed twist. If the tubes are stacked together to form a left-handed corner, the directors arrange themselves so that an s ˆ 1 defect line is formed. The singularity can be removed, however, since all integer defect lines can escape. But the directors for a right-handed corner form an s ˆ ÿ 12 defect. This singularity cannot be removed, since half-integer defect lines cannot escape. As we shall see, although the cubic blue phases can be described as lattices of double-twist tubes, frustration dictates that there also be an interpenetrating lattice of defects [12].

7.1.3

Defect-Mediated Melting

Defect theories of melting have been discussed in the context of crystal melting for some time [13]. In the usual three-dimensional defect melting theory, the free energy Fdef for production of a defect out of a perfect crystal is calculated as a function of temperature. As the temperature increases, Fdef is found to decrease smoothly from a positive to a negative value, passing

7. Blue Phases

189

Figure 7.2. Stacking of double-twist tubes and frustration. The twist in each tube is taken to be right-handed, twisting to 45 at the tube boundary. (A) Righthanded corner: the directors form an s ˆ ÿ 12 defect that cannot escape and is singular. (B) Left-handed corner: the directors form an s ˆ 1 defect which can escape and is nonsingular. From Sethna [9].

through zero at some temperature Tc . Thus, below Tc defects are energetically unfavorable and the crystal is stable. Above Tc , however, defects are energetically favorable, a catastrophic onset of defects occurs, and the resulting liquid can be thought of as a solid saturated with defects. Carrying this idea over to the helical±isotropic transition, there are two di¨erences. First, we must use disclinationsÐtopological line singularities in the director ®eld of the liquid crystalÐrather than crystal defects. The second di¨erence is that the helical phase, which has no defects, ``melts'' to the blue phase, which is characterized by a stable defect lattice of line disclinations rather than by a random collection of defects. Indeed, there is more than one way to create such a lattice: thus BPI and BPII. The helical phase therefore melts to BPI, BPI melts to BPII, and, with a ®nal onset of randomly positioned defects, BPII melts to the isotropic phase. An intermediate phase consisting of a defect lattice is not an ingredient of the ordinary solid±liquid defect-melting scenario. It comes about because of liquid crystalline order and, as we shall see, the presence of high chirality.

7.1.4

Goal of This Review

A number of reviews of blue phases have already been written, notably those by Stegemeyer and coworkers [14], [15], Belyakov and Dmitrienko [16], Crooker [8], [17], Cladis [18], Seideman [19], and Wright and Mermin [20]. These reviews cover the experimental and theoretical work until 1989 when the cubic blue phases were most actively investigated. Since that time, e¨ort

190

P.P. Crooker

has been mainly directed toward understanding BPIII, although the cubic blue phases have not been ignored. This chapter will therefore discuss the early work rather quickly, the goal being to get to the more recent work. In particular, the emphasis will be more on experiment than theory, since the earlier theoretical reviews are quite extensive and contain much more mathematical detail than we care to reproduce here. Finally, the number of papers in the ®eld has now grown quite large, and we despair of mentioning every work published. We will cover the main articles, however, and the reader should be able to obtain additional material from the references therein.

7.2

General Experimental Picture

Because the early blue phases were in cholesteryl esters and were di½cult to see in a microscope, there was early skepticism over whether these phases existed at all. Now, as a result of thermal and optical observations, that doubt has been laid aside.

7.2.1

Thermal Data

The existence of three thermodynamically distinct blue phases is now ®rmly established by thermal, optical, and viscoelastic measurements. The initial experiments showed density discontinuities [21] and new peaks in di¨erential scanning calorimetry traces [14], [22], [23], [24]. Additional evidence is found in viscoelastic data [25]. The most convincing proof, however, is revealed by heat capacity measurements [25], [26], [27]; those of Thoen [26] are shown in Figure 7.3. Here, in cholesteryl nonanoate, we see three peaks riding on the edge of the helical±isotropic peak; these peaks separate the helical, BPI, BPII, BPIII, and the isotropic phase. The sizes of the latent heats (in J molÿ1 ) are worth noting: Helical, 18; BPI, 5.8; BPII, 1.9; BPIII, 170; Iso. The value of 170 J molÿ1 is typical for the nematic±isotropic transition; the blue phase transitions are less than a twentieth of that value. One can then argue that the order in the blue phases is consequently closer to the helical phase than to the isotropic phase, but as we shall see, that is only true for BPI and BPII. One might also conclude that BPII and BPIII are very similar since the latent heat between them is so small. In fact, we shall see that BPIII has much more in common with the isotropic phase.

7.2.2

Phase Diagrams

The easiest way to observe blue phases, provided their selective re¯ection wavelengths are in the visible wavelength region, is by polarized re¯ection microscopy. The visual appearance of the various textures is given by Crooker [17]. One would then like to see how the blue phases appear as a

7. Blue Phases

191

Figure 7.3. Heat capacity Cp =R versus temperature T in cholesteryl nonanoate (CN). Helical phase (CH), blue phases (BPI, BPII, BPIII), isotropic phase (I) (from Thoen [26]).

function of chirality 2p=P, where P is the pitch. This can be achieved by mixing a chiral liquid crystal (enantiomer) with its racemate (a 50%±50% mixture of left- and right-handed enantiomers). Except for their chirality, the enantiomers are chemically identical and, to good approximation, may be mixed without changing the coe½cients of the Landau free energy. For weight percent X of the pure enantiomer in a chiral±racemic mixture, P ˆ P0 =X . Figure 7.4 shows the phase diagram of CE2 (EM Chemicals), one of the most chiral liquid crystals available. Similar diagrams [28], [29], [30], [31] have shown that the blue phases always appear in the sequence BPI, BPII, and BPIII as the chirality is increased, and that, while BPI and BPIII appear to be increasingly stable at high chirality, BPII only exists over a limited region. Although BPIII is di½cult to observe visually (the pitch of CE2 becomes so short that the selective re¯ections move into the ultraviolet), we shall discover later that the BPIII±isotropic coexistence line is ®rst order, ending in a critical point at a higher chirality.

7.3

BPI and BPIIÐTheory

Research on blue phases has always been characterized by a close interplay between experiment and theory. The initial experiments were reported around 1979; in 1980, Hornreich and Shtrikman published the ®rst Landau

192

P.P. Crooker

Figure 7.4. Transition temperatures T versus mole fraction chiral CE2 in a chiral± racemic mixture. The limited chirality range of BPII appears to be a universal feature of blue phases.

theory of a cubic blue phase [32]. This paper gave a tremendous boost to the ®eld (although, astonishing to this reviewer, the theory seemed more credible to some than the experiments themselves!). Since that time, however, the theory has pointed the way to new experiments, while the experiments have suggested new avenues for theory. Here we concentrate on theories of the cubic blue phasesÐBPI and BPII. Only the main theoretical ideas are given; for more detail, we refer the reader to in-depth theoretical reviews and the references therein.

7.3.1

Landau Theory

A Landau theory for the cubic blue phases was ®rst proposed by Brazovskii and Dmitriev [3], Hornreich and Shtrikman [32], [33], [34], [35], [36], [37], and Kleinert and Maki [38]. Detailed reviews of this approach have been given by Seideman [19], Belyakov and Dmitrienko [16], Wright and Mermin [20] and, in particular, by Hornreich and Shtrikman [39]. All Landau theories begin by ®rst identifying an order parameter which incorporates the symmetry of the lower phase and which becomes zero in the upper phase. Next, a Landau free energy is constructed from the lowest powers of the order parameter, taking care to retain all terms allowed by symmetry and having a leading coe½cient which is a linear function of temperature, passing through zero at some temperature T  . Finally, the free energy is minimized with respect to the order parameter to ®nd the structure. In the case of blue phases, however, the form of the free energy is su½ciently complicated that such a global minimization has not been possible. The

7. Blue Phases

193

strategy has therefore been to compare the free energy of the isotropic phase with that of several proposed alternative structures to determine which one is most stable. We ®rst sketch out the theory without mathematical detail in order to give a general picture of the process. Let the order parameter be given by a symmetric, traceless tensor Q…r† which will just be the anisotropic part of the dielectric tensor. Since the Landau free energy is a scalar quantity, we expand it in lowest orders of scalar combinations of Q. In very schematic form, this is just … 1 …7:1† ‰aQ 2 ‡ c…`Q† 2 ÿ d…`  Q†
Q ÿ bQ 3 ‡ gQ 4 Š d 3 r; Fˆ 2 where a ˆ a…T ÿ T  † and a and the rest of the coe½cients are constants. The coe½cient d of the chiral term is proportional to the chirality. The terms of order Q n give us the usual ®rst-order nematic±isotropic transition; the …`Q† 2 term represents the gradients in Q necessary to produce a cubic structure; and the Q
…`  Q† term is a chiral term allowed because the molecules themselves lack mirror symmetry. It is this chiral term which greatly complicates the situation and causes the appearance of blue phases. Since the anticipated cubic structure is periodic, we next expand Q…r† in a Fourier series with wave vector k: Q…r† ˆ

2 X X

‰Qm …k†Mm …k†Še i…k
r‡cm …k†† ;

…7:2†

k mˆÿ2

which says that each Fourier component of wave vector k has a Fourier coe½cient given by the quantity in brackets and a phase given by cm …k†. The coe½cient has two parts. Mm …k† is the tensor part (because Q is a tensor) which gives the director symmetry of that particular Fourier component. Because the director symmetry is just that of the spherical harmonics Y2 m …y; f†, the Mm …k† tensors are represented by the Y2 m Ðone for each of the ®ve values m ˆ 0; G1; G2. The other part of the coe½cient is a scalar, Qm …k†, which represents the amplitude or strength of that particular Fourier component with symmetry Mm …k†. The sum over m includes all the possible m values for a particular k, while the sum over k includes all those wave vectors allowed by the symmetry of the system. In the isotropic phase, all k values are allowed. For the blue phase, however, one ®rst chooses a particular unit cell to calculate, then allows only those values of k, along with the accompanying phase factors cm …k†, corresponding to the reciprocal lattice vectors …hkl† for that unit cell. For example, if the only nonzero coe½cients Qm …k† are those corresponding to k vectors lying along the body-centered cubic (bcc) (110) directions, one ®nds that a structure with the bcc O 5 space group symmetry appears between the helical and isotropic phases. Allowing other coe½cients to be nonzero will allow other structures to be tested which may or may not be more stable.

194

P.P. Crooker

Substituting (7.2) in (7.1), one then adjusts the coe½cients Qm …k† in order to minimize the free energy for the particular structure. A comparison of the free energies for this and other structures for various temperatures and chiralities then allows the construction of a temperature±chirality phase diagram. Putting in some of the mathematical details [39], the Landau theory for the chiral nematic±isotropic transition has been described by de Gennes [2], who utilizes a tensor order parameter which is just the anisotropic part eij …r† of the total dielectric tensor eijd …r†: eij …r† ˆ eijd …r† ÿ 13 Tr…eij †dij :

…7:3†

This tensor is symmetric, traceless, and will vanish in the isotropic phase. The free energy includes the lowest-order scalar combinations of these tensors and their derivatives: … 2 2 2 2 d 3 r‰12 …aeij2 ÿ 2deijl ein ejn; l ‡ c1 eij; Fˆ l ‡ c2 eij; i † ÿ beij ejl eli ‡ g…eij † Š: …7:4† V

In this expression, the coe½cients d; c1 ; c2 ; b, and g are constants, while a ˆ a…T ÿ T  † changes sign when T ˆ T  . The three terms with coe½cients a; b, and g are just those terms which appear in the nematic free energy; the c1 and c2 terms are order parameter gradients …eij; l ˆ qeij =qxl † which are also allowed for nematics. The chiral term with coe½cient d is forbidden in nematicsÐit contains the antisymmetric tensor eijl and lacks nematic mirror symmetry. This free energy is su½cient to treat the helical±isotropic transition, but due to the cubic term, global minimization is extremely di½cult. In fact, there is no guarantee that the helical phase is the lowest energy phase available to nature. Hornreich and Shtrikman therefore propose other possibilitiesÐcubic phasesÐand by comparing the resulting energies with the helical and isotropic energies are able to determine the most stable structure for given temperature and chirality. In order to describe cubic phases, eij is expanded in Fourier components as usual; but rather than allow all wave vectors, Hornreich and Shtrikman limit the expansion to just those wave vectors which correspond to the reciprocal lattice vectors of particular cubic space groups. The Fourier transform of eij …r† then takes the form X ÿ1=2 Nhkl eij …h; k; l†e ikhkl …hx1 ‡ kx2 ‡lx3 † : …7:5† eij …r† ˆ h; k; l

Thus a particular cubic structure is characterized by reciprocal lattice vectors khkl with Miller indices …h; k; l†, and there are Nhkl reciprocal lattice vectors of length khkl . The eij …h; k; l† tensor consists of the Cartesian representations of the spherical harmonics Y2m …y; f†. Schematically,

7. Blue Phases

eij …h; k; l† ˆ

2 X

em; hkl e icm; hkl ‰Mm; hkl Š

mˆÿ2

0

ÿ1

B ˆ e0 e ic0 @ 0

0 0 ‡ e1 e

ic1 B

0 0

0

0

1 1

B ‡ e2 e ic2 @ i

0

i

1

1

2 1

C i A ‡ c:c:

0 i

ÿ1 0

…7:6†

C 0A

ÿ1

@0 0 0

0

195

0

1

C 0 A ‡ c:c:

…7:7†

0

Here c.c. means complex conjugate; also we have left out some numerical coe½cients and suppressed the indices hkl on em ; cm , and Mm . The basis matrices are de®ned in a local right-handed coordinate system for each khkl . Thus, for any particular wave vector khkl there may be as many as ®ve different Fourier components …m ˆ 0; G1; G2†, each having a director symmetry speci®ed by Mm , an amplitude given by em , and a phase by cm . The remaining strategy is then to decide on a particular space group to describe, write down the appropriate Fourier components and their phases ab initio, then determine the magnitude of the coe½cients by numerical minimization of the free energy. This process leads to cubic structures through the following reasoning: Since the minimization involves the cubic term, one seeks contributions from wave-vector triads whose vector sum is zero, that is, a triangle of wave vectors. Selection of wave vectors with m ˆ 2 symmetries will also minimize the quadratic part of the free energy. The simplest three-dimensional structure is obtained by choosing a set of wave vectors forming the six edges of a regular tetrahedron. The result is a bodycentered cubic (bcc) structure with the space group O 5 (I432). Utilizing this basic tetrahedron of wave vectors and adding harmonics, Hornreich and Shtrikman have been able to construct other bcc structures, along with simple cubic (sc) and face-centered cubic (fcc) structures. The next step is therefore to construct a phase diagram showing that particular structure which minimizes the free energy at each point of the temperature± chirality plane. An example of such a phase diagram is given in Figure 7.5, which shows temperature t and chirality k in nondimensional, reduced units. The various phases represented are the usual isotropic (I) and helical (C) phases, plus cubic blue phases, with space groups given by bcc O 5 (I432), bcc O 8 …I 41 32†, and sc O 2 …P42 32†. The big success of this theory is that cubic phases are shown to indeed be more stable than the helical phase at the helical±isotropic boundary.

196

P.P. Crooker Figure 7.5. Theoretical phase diagram from the Landau theory with higher harmonics. Temperature t and chirality k are in reduced units. The helical (C), isotropic (I), and cubic …O 2 ; O 5 ; O 8 † phases are allowed. From Grebel et al. [37]. This ®gure should be compared with Figure 7.4.

Furthermore, both bcc and sc structures have been found, which agrees with experimental ®ndings. Comparison of Figure 7.5 with Figure 7.4, however, reveals that the experimental and theoretical phase diagrams di¨er considerably in detailÐnot only quantitatively, but qualitatively as well. Finally, the analysis presented here only predicts the cubic BPI and BPII phases. Despite attempts, the noncubic BPIII phase does not come out of the analysis. Extensions of the Landau theory are possible, for example, by introducing higher-order elastic terms in (7.2) as has been done by Longa et al. [40]. The resulting phase diagrams are richer, but still di¨er from the experimental phase diagrams.

7.3.2

Defect Theory

The fundamental cause of macroscopic chirality in liquid crystals is that the molecules themselves are chiral. Consequently, the interactions between adjacent molecules are also chiral: twist arises because the interaction energy of two adjacent molecules is minimized when they are at a slight angle to each other. In the conventional helical phase, this condition is only partially metÐtwist occurs along the single axis k but not along axes perpendicular to the twist axis. Clearly, the energy can be further lowered by allowing twist in all directions perpendicular to the local director. Such a con®guration, called double twist, is shown in Figure 7.6. It has been shown that a tube of double twist does indeed have lower free energy than the conventional, single-twist helical structure, but only near the axis [1], [9]. If the local twist becomes as much as 90 , substantial bend distortion occurs and the energy advantage is lost. Consequently, the twist in a tube is limited to 45 . One can then envision assembling double-twist tubes in a regular lattice having the space group symmetries discussed above, ®lling the interstices with nematic material, and allowing the directors to relax everywhere to achieve equilibrium. If such a program is carried out, the intersections between double-twist tubes can be shown to have incompatible directors, which leads in turn to frustration and the introduction of disclinations, as described earlier [9].

7. Blue Phases

197

Figure 7.6. Cross-sectional view of a double-twist region. The director is parallel to the tube axis at the center, twisting along any radius. The energy of such a con®guration is lower than for twist in a single direction, but only near the center. In a double-twist tube, the angle at the edge is A45 .

One can think of the blue phase as a lattice of double-twist tubes (which necessitates a lattice of disclinations) or a lattice of disclinations (which necessitates a lattice of double-twist tubes) [20]. Thus, a theory involving a lattice of double-twist tubes becomes implicitly a theory for a lattice of defects. The defect theory of blue phases was initially introduced by Meiboom, Sethna, Anderson, and Brinkman [5], [6], [12], [41], [42], [43]. For detailed reviews see Sethna [9] and Wright and Mermin [20]. Calculating the free energy for a lattice of defects is a di½cult task. Meiboom et al. [12] start by writing the free energy per unit length Fdisc of a single disclination line, which consists of four terms: Fdisc ˆ Fel ‡ Fsurf ‡ Fcore ‡ Fint :

…7:8†

The ®rst term is the elastic energy associated with the defect, which has the form Fel @ K ln…R0 =Rc †:

…7:9†

This term comes from calculating the usual Frank free energy (i.e., with bend, twist, and splay, and equal elastic constants) outside a disclination core of radius Rc but inside a cuto¨ radius size R0 . The second term is an additional elastic term … 1 K`
‰…n
`†n ÿ n…`
n†Š d 3 r Fsurf ˆ 2 @ ÿK;

…7:10†

which is usually dropped because it transforms to a surface integral. In this case, however, the integral must also be taken over the inner surface of the defect core and cannot be ignored. This is, in fact, the term that pulls the energy of the defect below zero.

198

P.P. Crooker

The third term Fcore ˆ a…Tiso ÿ T†pR02 ;

…7:11†

arises because the energy cost of maintaining a highly strained, nonzero order parameter at the core is greater than the energy required to drive the core itself isotropic, even though the temperature is below the usual helical± isotropic transition temperature Tiso . The quantity a…Tiso ÿ T† is just the di¨erence between the free energies of the isotropic and helical phases. Finally, the last term Fint ˆ 2psRc ;

…7:12†

is an interfacial energy, characterized by surface tension s, between the core and the chiral material outside. Its importance is probably minor. Summing all these terms, it is easy to see that the free energy is positive for T far enough below Tiso , but may become negative for temperatures close to, but still less than, Tiso . This feature opens the possibility for a transition to a blue phase lattice just below Tiso . The strategy is to assemble a lattice of defects having a particular space group symmetry and ®ll the interstices with nematic material. The director is then allowed to relax everywhere (except of course at the defects) and the free energy calculated. From this process one ®nds that there is also an interpenetrating lattice of double-twist tubes and that the whole structure is stable between the helical and isotropic phases. Figure 7.7 shows both the double-twist lattice and the defect lattices for the proposed structures sc O 2 and bcc O 8ÿ . Other structuresÐbcc O 5 and bcc O 8‡ Ðhave also been worked out [42].

7.3.3

Comparison of the Landau and Defect Theories

The Landau theory uses an order parameter with cubic symmetry which is relaxed over a cubic unit cell in order to determine the minimum free energies of various structures. The defect theory utilizes an assumed lattice of defects and allows director orientations to conform to this lattice. Wright and Mermin [20] have argued that the Landau theory is therefore a ``high chirality'' model, while the defect theory describes ``low chirality.'' Despite these di¨erences, however, the Landau and defect theories have many similarities. Both theories result in double-twist regions, which are ®tted together to form three-dimensional structures. Both theories also include defectsÐin the defect theory they are explicit disclination lines where the order parameter goes to zero abruptly; in the Landau theory, they appear implicitly as regions where the order parameter becomes zero gradually. To describe these defects in Fourier space, the defect theory would require many higher-order, spatial Fourier components. The Landau theory, on the other hand, uses very few Fourier components. Thus, the main di¨erence between the Landau and defect theories may just be the number of Fourier compo-

7. Blue Phases

199

Figure 7.7. Models of cubic blue phases. (a) Arrangement of double-twist tubes for the sc O 2 structure. (b) Corresponding unit cell of defect lines for the sc O 2 structure. (c) Arrangement of double-twist tubes for the bcc O 8ÿ structure. (d) Unit cell of defect lines for the bcc O 8ÿ structure. From Dubois-Violette and Pansu [10].

nents included in the order parameter rather than any fundamental di¨erences in the structures. For excellent discussions of the blue phases from the viewpoint of geometry, topology, and frustration, the articles by Dubois-Violette and Pansu [10] and Pansu [44] are recommended.

7.3.4

The Bond-Orientational Order and Fluctuation Models

In any theoretical model of the cubic blue phases, a meeting point with experiment should be a comparison of theoretical and experimental phase diagrams. Despite theoretical successes, however, the theoretical phase diagrams (Figure 7.5) are still missing essential features of the experimental phase diagrams (Figure 7.4). The missing features are:

. Theory has correctly predicted the space groups of the helical, BPI, and

BPII phases, and their order with increasing temperature is theoretically correct. The theoretically predicted bcc O 5 phase, however, has never been detected experimentally. . In the theoretical phase diagrams, the BPII region becomes broader with increasing chirality. Experimentally, however, BPII vanishes at high chirality. . The theoretical phase diagrams do not reproduce the experimentally observed BPIII phase.

200

P.P. Crooker

. The theoretical phase diagrams do not reproduce the experimentally observed critical point which terminates the BPIII±isotropic coexistence line at high chirality.

The cubic bond-orientational order (BOO) model of Trebin and coworkers [45], [46] is an attempt to rectify these theoretical problems. Originally formulated for crystals, the central idea of BOO is that the positional order of the atoms can be lost (due to formation of defect pairs or ¯uctuations) while still retaining the orientational order of the bonds. An analogous situation would occur for a ¯oor loosely tiled with square tiles. If the sides of the tiles are all aligned properly, but the tiles are not ®tted together correctly, orientational order is retained while positional order is lost. In the blue phases, the atoms and bonds are replaced by the unit cell's corners and edges, respectively. The goal of the BOO model of blue phases is to convert the higher temperature O 5 phase to a cubic phase with only bondorientational order, which might then be the amorphous-appearing BPIII. The authors accomplish this by adding a ¯uctuational term to the usual Landau free energy, the minimization of which leads to various phase diagrams depending on the coupling strength. These phase diagrams have the required BOO phase just below the isotropic phase, but the O 5 itself is not completely destabilized. Nevertheless, the authors claim that the BOO phase would be a likely candidate for BPIII. With the realization that the BPIII±Iso transition ended in a critical point, Trebin and coworkers [47] have also proposed a ¯uctuation-dominated model of the blue phases. Here the order parameter coe½cient is separated into a non¯uctuating term and a ¯uctuating term and the free energy accordingly separated into a mean-®eld and a ¯uctuating part. On the one hand, the resulting phase diagrams are encouraging: the O 5 phase does not appear (but only for large transition temperatures) and BPII disappears at high chirality. On the other hand, a simpli®cation of the theory provides only ``a hint of the occurrence of a second isotropic phase [i.e., BPIII] and of a critical point at high chiralities'' [47].

7.4

BPI and BPIIÐExperiment

A number of experimental techniques have been used to elucidate the structure of BPI and BPII. In what follows, we will summarize them and show how the accumulated experimental evidence gives us a good picture of these phases.

7.4.1

Bragg Scattering

It is well known that the helical phase with a planar texture will selectively re¯ect light. Selective re¯ection requires light which has a circularly polar-

7. Blue Phases

201

Figure 7.8. Selective re¯ection wavelengths lSR versus temperature for a mixture of chiral CB15 and nematic E9. Note that BPI supercools into the helical (Ch) phase, whereas the BPI±BPII transition is reversible. The ratios of the wavelengths are those for simple cubic or body-centered cubic lattice symetry (from Johnson, Flack, and Crooker [54]).

ized component with the same handedness as the helix and a wavelength satisfying l ˆ nP, where n is an average refractive index and P is the pitch. In the cubic blue phases, however, there may be several selective re¯ection wavelengths corresponding to various crystal planes, none of which coincide with the helical selective re¯ection wavelength. Initial reporting of this e¨ect was by Stegemeyer and coworkers [14], [48], [49], [50], [51]. In a related experiment, Meiboom and coworkers [42], [52], [53] measured the transmitted light through an unaligned blue phase sample. Such a measurement yields a step in the transmitted intensity at each selective re¯ection wavelength. The selective re¯ection wavelengths of a mixture of chiral CB15 and nematic E9 (both obtained from BDH Chemicals) are shown in Figure [54]. p 7.8 p  In both BPI and BPII, the wavelengths have the ratios l0 , l0 = 2; l0 = 3; . . . ; where l0 is the longest selective re¯ection wavelength. These ratios are well known from X-ray di¨raction: they are the signature of Bragg scattering from either a body-centered cubic (bcc) or simple cubic (sc) lattice. The conclusion, then, is that BPI and BPII are cubic, but with lattice parameters of the order of visible light wavelengths. Furthermore, as can be seen in Figure 7.8, the lattice parameters of BPI and BPII are di¨erent. One then wonders whether the lattice structures are also di¨erent. To distinguish bcc from sc, Hornreich and Shtrikman [55], [56] and Belyakov et al. [57] worked out the polarization selection rules for each possible sc and bcc space group. As shown in (7.7), each set of Bragg planes contains ®ve order parameter coe½cientsÐe0 ; eG1 , and eG2 Ðeach of which

202

P.P. Crooker

re¯ects circularly polarized light (rcp or lcp polarizations) in a unique way. In particular, the third line of the sc lattice ‰sc …111†Š should re¯ect circularly polarized light as an ordinary mirror …rcp ! lcp; lcp ! rcp†, whereas the third bcc line ‰bcc …211†Š should re¯ect only that polarization corresponding to the twist of the helix …rcp ! rcp; lcp ! no reflection†. However, experiments relying on this technique [58] give results which violate these rules and are in con¯ict with other methods, namely platelet growth morphology and Kossel diagrams. Despite e¨orts to resolve the selection rule contradiction [59], [60], the latter techniques have proved a more reliable way of identifying structures. Another way to identify the dominant contributions of the order parameter for a particular line, is to experimentally determine the re¯ecting Mueller matrix of the Bragg-re¯ecting blue phase. The polarizations of light incident and scattered from a surface can be described by four-component Stokes vectors. The matrix which transforms one to the other (and describes the blue phase) is a 4  4 Mueller matrix. By analyzing the re¯ected Stokes vectors for a range of incident Stokes vectors, the complete Mueller matrix can be determined. The result of such measurements [61], [62] was that the eÿ2 coe½cient completely dominates the other coe½cients, which rules out certain space groups.

7.4.2

Kossel Diagrams

A specialized case of Bragg scattering is the Kossel diagram technique, ®rst used for blue phase analysis by Pieranski and coworkers [63], [64], [65], [66] and more recently by Miller and Gleeson [67], [68]. Figure 7.9 illustrates the basic principle [68]. The blue phase sample S, with one of the sets of crystal planes shown, is illuminated with highly convergent light through microscope objective L. If the light is monochromatic with wavelength l=n in a material with crystal planes separated by distance d, a particular cone of the incident rays will satisfy the Bragg condition l ˆ 2nd sin…y†. This cone of rays is backscattered, refocused through the objective, and brought to an image in the back focal plane FP of the objective. Examination of this plane reveals the Kossel diagram, in which each of the sets of crystallographic planes in the sample is projected on the back focal plane as a circle or an ellipse. Without further analysis, it is clear that the symmetry of the Kossel diagram is just the symmetry of the crystal itself. Thus, the longest wavelength bcc line [bcc (110)] and the longest wavelength sc line [sc (100)] can be distinguished because only the sc (100) Kossel line has fourfold symmetry. In addition, quantitative measurements of the Kossel diagrams allows the angles and crystal-plane spacing to be determined. This method has proven to be of great utility in characterizing the electric-®eld-induced BPX phase and in determining that the symmetry of BPII is sc O 2 …P42 32†. More recently, Kossel diagrams have been utilized by Miller, Gleeson, and Lydon [69] to address the well-known ``phase problem''Ðthat is, to

7. Blue Phases

(a)

203

(b)

Figure 7.9. Principle of the Kossel diagram technique. (a) From the cone of incident monochromatic light, sets of crystal planes in sample S Bragg scatter light back through the objective lens L to make a Kossel diagram at focal plane FP. (b) Each set of planes is represented by an arc in the Kossel diagram (Kossel diagram courtesy of B. JeÂroÃme).

determine the relative phases of the Bragg re¯ections. This phase determination is achieved by examining the regions where two Kossel rings either intersect or pass close to each other. Interference fringes are observed experimentally, and when these fringes are modeled by a simple theory, they are seen to depend sensitively on the phase shift between the lattice planes causing the respective rings. Figure 7.10 shows experimental and theoretical results from two Kossel rings intersecting perpendicularly; note the presence of gaps in the lines near the intersection. In particular, the (100) and (010) planes of BPII have been examined, which, according to the double-twist

Figure 7.10. Interference patterns for Kossel lines intersecting at right angles. Left: experiment. Right: theory (from Miller et al. [69]).

204

P.P. Crooker

model (see Figure 7.7), should have a relative phase shift of p. The data do not account for this phase shift, however, and hence details of the BPII structure remain unresolved.

7.4.3

Crystallite Morphology

Much of the original information on blue phases was deduced from microscopic examination of the textures [70], [71]. Electron microscopy of freezefractured samples has also been performed, but the results are di½cult to interpret [72], [73]. A more reliable and direct way to determine a crystal's space group is to carefully grow the crystal and observe its shape, which has the same symmetry as the unit cell. It is possible to do this in the two-phase region of a multicomponent mixture, and/or in a temperature gradient. Contributors to this e¨ort have been primarily Stegemeyer and coworkers [74], [75], [76], [77], [78] and Pieranski and coworkers [79], [80], [81], [82]. See also the review by Stegemeyer [15]. Figure 7.11 is an example of such a crystallite in BPI. From pictures like these, along with Kossel diagram information, it has been determined that BPI is body-centered cubic with space group O 8 …I 41 32†, and BPII is simple cubic with space group O 2 …P42 32†. Both of these space groups are shown to be stable in the Hornreich and Shtrikman [39] theory.

Figure 7.11. BPI crystallite. Top: Microscope picture, viewed along the (200) axis. Bottom: From left to right, views along the (110), (211), and (200) axes (from BarbetMassin, Cladis, and Pieranski [79]).

7. Blue Phases

7.4.4

205

Rotatory Power

The experimental signature of a chiral structure is rotatory power, in which the plane of polarization of a beam of linearly polarized light is rotated after passing through a chiral sample. The earliest rotatory power measurements established the chiral nature of the BPI and BPII blue phase peaks [48], [49], [83]. Later measurements have followed and have even been able to provide values of the order parameters [84], [85], [86], [87]. Theories for the behavior have been provided by Bensimon et al. [88] and Belyakov et al. [84]. The biggest utility of rotatory power, however, has been in elucidating the nature of BPIII, as we shall see below.

7.4.5

Viscoelastic Measurements

An important question to ask of blue phases is whether they are liquid or solid. The cubic structure is not one of atoms in ®xed positions as in a conventional crystalÐrather, the molecules are free to di¨use randomly throughout the blue phase lattice, changing their orientation en route so as to conform to the blue phase's spatially dependent order. Solids are characterized by a nonzero static shear elastic constant, whereas a liquid will not support static shear. Conventional viscometric techniques (i.e., capillary ¯ow and falling balls), which now seem rather crude considering the fragile nature of the blue phase lattice, initially showed a large viscosity peak at the helical±isotropic transition [89], [90], [91]. Viscosities and other transport coe½cients associated with the pretransitional region of the isotropic phase were addressed by light scattering [92], [93], [94], [95]. Once the cubic nature of the blue phase was established, attempts to measure the elastic constants using more sensitive techniques appeared shortly thereafter [25], [96], [97], with those of Kleiman et al. [25] being the most extensive. The latter experiments are very delicate, since the blue phase lattice is both soft (small elastic constants) and weak (small elastic limit). Torsional oscillators con®gured as cup viscometers were used and the shear distortion was kept to less than 0.02%. Figure 7.12 shows results for both the shear elasticity G and the viscosity h. These data are taken at various frequencies and must be extrapolated to 0 Hz to obtain the static properties. In the helical phase the extrapolation is somewhat dependent on the model; nevertheless, the authors claim that G becomes nearly zero in the helical phase and about 710 dyn cmÿ2 in BPI. (This ®gure should be compared to @10 11 dyn cmÿ2 in a metal!) However, since BPI also possesses viscosity, its behavior is that of a viscoelastic solid. Another way to assess the elastic constants in the blue phases is to observe their behavior in the Cano wedge con®guration, which tends to compress and stretch the blue phase lattice parameter. This strain also raises the free energy and alters the blue phase transition temperatures, as has been shown by Feldman et al. [98]. For more discussion of blue phases in the Cano wedge con®guration, see the papers by Stegemeyer and coworkers [99], [100].

206

P.P. Crooker Figure 7.12. (a) Shear elastic constant and (b) viscosity for cholesteryl nonanoate for di¨erent frequencies (from Kleiman et al. [25]).

7.5

Electric Field E¨ects

When electric ®elds are applied to liquid crystals, the molecules tend to alignÐeither parallel to the ®eld (for ea > 0) or perpendicular (for ea < 0). For the case of nematics, which already have a preferred direction, the director is simply reoriented without breaking the symmetry. However, the helical phase has two nonequivalent directions: the twist axis, and the director, which rotates spatially about the twist axis. If ea > 0, such a helical director is clearly incompatible with a uniform ®eld. For this case, an increasing ®eld ®rst distorts the helix, then stretches out the pitch, and ®nally causes the well-known cholesteric±nematic transition [1]. If ea < 0, the helical director is only compatible with a uniform ®eld if the twist axis and ®eld are parallel. Since the cubic blue phases have three equivalent axes, an applied ®eld breaks the cubic symmetry and creates a preferred axis. But, like the helical phase, blue phases are chiral, being composed of a lattice of double-twist tubes. It is therefore not surprising that applied ®elds lead to distortion of the lattice (electrostriction) and, for high enough ®elds, new lower-symmetry phases. These e¨ects occur for both ea < 0 and ea > 0. Electric ®eld e¨ects on blue phases have been reviewed previously by

7. Blue Phases

207

Kitzerow [101]. Since electric ®eld-induced phases were studied ®rst, we describe them next. Electrostriction of blue phases will then be discussed at the end of this section.

7.5.1

Electric Field-Induced Phases

The ®rst experiments with electric ®elds showed that an increasing ®eld initially lengthens the blue phase lattice parameter and causes birefringence. A theory for weak ®elds was given by Lubin and Hornreich [102]. With larger ®elds, the blue phases may transform between themselves, to the helical phase, and ultimately to the nematic phase [23], [103], [104], [105], [106], [107], [108], [109]. Fields also a¨ect the orientation and facetting of blue phase crystallites [81], [82], [110]. At the same time, Hornreich, Kugler, and Shtrikman [111], [112] predicted from Landau theory that a two-dimensional hexagonal blue phase, BPH 2D , should be stabilized by an electric ®eld when the dielectric anisotropy is positive …ea > 0†. This prediction led to a search for other blue phase structures. Subsequent temperature±electric ®eld phase diagrams [113], [114] showed that other phases did in fact occur, but identi®cation of the new structures was much more di½cult. Figure 7.13, due to Porsch and Stegemeyer [113], shows a CB15/E9 mixture in which a new phase, BPE, is detected. Figure 7.14, due to Pieranski et al. [115], shows the same material and identi®es hexagonal platelets found by growing crystallites in the BPII±isotropic twophase region. The presence of hexagonally shaped crystallites was the ®rst reliably identi®able hexagonal phase.

Figure 7.13. Voltage±temperature phase diagram for a 49.6% mixture of chiral CB15 nematic E9. The helical phase is labeled ``chol''; the structure BPE was not yet determined in this article (from Porsch and Stegemeyer [113]).

208

P.P. Crooker Figure 7.14. Schematic voltage±temperature phase diagram for a 49.8% mixture of CB15 in E9. The shaded region is the coexistence region in which crystals of di¨erent shapes appear. C is the helical phase; H is the hexagonal BPH 3D phase (from Pieranski et al. [115]).

This hexagonal phase was not, however, the predicted BPH 2D phase. Since the hexagonal crystallites exhibited circularly polarized Bragg re¯ections, it was evident that the observed phase was three-dimensionalÐnow identi®ed as BPH 3D [D64 …P62 22† or D65 …P64 22†]. Later work [116] has shown that increasing the electric ®eld causes the re¯ected intensity to weaken and ®nally, at a threshold ®eld, to disappear altogether. The platelets remain, however; they are in the BPH 2D phase and are detectable in transmission. Decreasing the electric ®eld causes the colored hexagonal BPH 3D platelets to reappear. As it turns out, positive dielectric anisotropy is not a requirement: experiments [117] have found three-dimensional hexagonal phases in systems with ea < 0. Later theoretical work by Hornreich and Shtrikman [118], [119] has included the possibility of two- and three-dimensional hexagonal structures for both positive and negative dielectric constant. For ea > 0, in addition to their previously predicted BPH 2D they predict two possible three-dimensional hexagonal phases which they call BPHa3D and BPHb3D . For ea < 0, BPH 2D a two-dimensional phase does not appear, but a single three-dimensional phase, BPHc3D , does. The results of their calculation are shown in Figure 7.15. So far only one three-dimensional hexagonal phase has been identi®ed for each sign of ea . A second hexagonal phase for ea > 0 is either not present or has escaped detetection. Further investigation on the same system [64], [120] has revealed a second ®eld-stabilized phase, called BPX, which has a body-centered tetragonal unit cell with space group D410 …I 41 22†. This structure can be grown by applying an electric ®eld along the BPI (100) direction, as shown in Figure 7.16. From Kossel diagrams, it is evident that the cubic unit cell ®rst distorts, becoming orthorhombic. Then, above a critical ®eld, the body-centered tetragonal phase locks in. Further details of the transition and of the resulting lattices are found by measurements of the Kossel diagrams. For a discussion of the geometry of BPX, see Pansu [44].

7. Blue Phases

209

Figure 7.15. Theoretical phase diagram showing temperature t versus squared electric ®eld e 2 in reduced units. The phase diagram is for reduced chirality k ˆ 1:7; results to the left (right) of the dashed line correspond to ea < 0 …ea > 0†. Shown are the helical phases (Cÿ or C‡ ), nematic phase (N), simple cubic phase …O 2 †, twodimensional hexagonal phase …BPH 2D †, and three-dimensional hexagonal phases …Ha3D ; Hb3D ; Hc3D † (from Hornreich and Shtrikman [119]).

7.5.2

Electrostriction

For ®elds which are too small to cause a phase transition, there are two e¨ects. The simplest is to favor a particular orientation of the crystallites without lattice distortion [82]. The second is electrostriction, namely a distortion of the cubic blue phase lattice due to applied ®elds. As in the previous discussion of elastic measurements, this distortion is fundamentally di¨erent from that which occurs in a conventional crystal where the lattice is composed of atoms at speci®c lattice points. In blue phases, it is the lattice of director orientations which does the distorting; the molecules themselves continue to be free to di¨use through this lattice. A phenomenological theoretical framework for electrostriction in blue phases was ®rst provided by Dmitrienko [121], with additional development

Figure 7.16. Transformation of the BPI unit cell into the tetragonal unit cell. From Kitzerow [101].

210

P.P. Crooker

by Trebin and coworkers [122], [123], [124]. In order to sketch the main ideas, we present this theory ®rst in a simpli®ed, one-dimensional form. Deformation of a solid object is given by the well-known strain tensor of elasticity theory. Let the point originally at location x be displaced by a small amount x. The strain tensor uij is then given by   1 qxi qxj ‡ : …7:13† uij ˆ 2 qxj qxi In order to produce an easily visualized situation, let us suppose that an internal electric ®eld Ex ˆ E produces a compression (or elongation) xx ˆ x in the x-direction only. Then uxx ˆ qx=qx ˆ u is the resulting strain and the free energy resulting from this process is [121]: F ˆ F0 ‡ 12 Lu 2 ÿ 12 eE 2 ;

…7:14†

where L is an elastic coe½cient and e is the dielectric permittivity. But e depends on the strain u and the ®eld E itself, which we describe by expanding e to lowest relevant orders in u and E: e ˆ e0 ‡ bu ‡ wE 2 :

…7:15†

The ®rst term, e0 , represents the unperturbed part of the permittivity; the second term describes the coupling of e to u, with b an elasto-optic coe½cient; and the third term describes higher-order terms in E, with w the nonlinear dielectric susceptibility. Substituting (7.15) into (7.14) and retaining only those terms containing u, one gets F ˆ F0 ‡ 12 Lu 2 ‡ 12 buE 2 :

…7:16†

Minimizing this equation with respect to u, we can ®nd the strain in terms of the electric ®eld and the phenomenological parameters L and b: u ˆ 12 lÿ1 bE 2 ˆ RE 2 ;

…7:17†

where R ˆ …Lÿ1 b†=2 is the electrostriction coe½cient. Thus, in our onedimensional example, the compression produced on the lattice by an electric ®eld in the same direction is proportional to E 2 . The electrostriction coe½cient R is the constant of proportionality and its sign determines whether the distortion is compression or elongation. Experiments can produce values for R; it is the task of theory to calculate R from more fundamental quantities. Of course, an electric ®eld in the x-direction may also cause distortion in the y- and z-directions, therefore a complete theory must take into account the tensor nature of the interactions. This is accomplished by replacing the quantities E; u; e; b; L; w, and R with, respectively, Ei , uij , eij , bijkl , Lijkl , wijkl , and Rijkl , and by replacing (7.17) with uij ˆ 12 lÿ1 ijkl bklmp Em Ep ˆ Rijmp Em Ep ;

…7:18†

7. Blue Phases

211

Figure 7.17. Bragg wavelengths l for various crystal planes versus electric ®eld E in BPI and BPII. (a) Negative dielectric anisotropy; (b) positive dielectric anisotropy (from Kitzerow [101]).

where repeated subscripts are summed over. Invoking the symmetry of the lattice greatly simpli®es the picture, however, and in the cubic case there are only three Rijkl coe½cients: R1111 ˆ R1 , R1122 ˆ R 2 , and R2323 ˆ R3 =2. Electrostriction experiments have been carried out for both BPI and BPII under conditions of both positive and negative dielectric anisotropy [105], [108], [125], [126], [127] using Bragg re¯ection and Kossel diagram techniques. The essential results are shown in Figure 7.17. Typically, the electrostriction coe½cients are @10ÿ15 ±10ÿ14 m 2 Vÿ2 and, as predicted theoretically, change sign if the dielectric permittivity changes sign. When the volume of the unit cell is unchanged, R1 ˆ ÿ2R 2 , as found experimentally [126], [127]. The value R1 =R3 is positive in BPII, but in BPI is negative and is called anomalous electrostriction [122]. Starting with the Landau theory, Trebin and coworkers [122], [124] have calculated the electrostriction coe½cients by allowing both the wave vectors and scalar amplitudes of the Fourier components to distort. Good agreement with the data is achieved, except for the case of anomalous electrostriction in BPI. The authors therefore conclude that the explanation for this behavior is beyond the capability of the Landau theory. As described earlier, the same group has also proposed a model of the blue phases incorporating bond-orientational order [45], [46]. However, a calculation of the anomalous electrostriction from this model [123] has had only limited success. Attention has also been paid to the dynamics of electrostriction. Experiments [128], [129], [130], [131] have shown that, to within instrumental accu-

212

P.P. Crooker

racy, the relaxation is single exponential, with relaxation times t @ 10 sÐ much longer than refractive index changes in the same material. These times have been ®tted to the phenomenological expression [130]: gL 2 ; …7:19† K where g is a viscosity, K is a Frank elastic constant, and the sample size L may be either the thickness of a cell-con®ned sample or the size of a crystallite. The L 2 behavior of t has been tested: agreement is fairly good for cell-con®ned samples but rather poor for crystallites. t@

7.6

BPIII

We now turn to BPIII, which has been the most enigmatic of the blue phases. Called variously the grey phase [14], the fog phase [70], the blue fog [103], and BPIII [53], the last name seems to have survived. This phase, which is amorphous and not cubic, has been reviewed in preliminary fashion by Crooker [17], Seideman [19], and Wright and Mermin [20], but at the time of these reviews (1989±1990) only the initial experiments had been performed and theoretical attention was just beginning.

7.6.1

Experiment

A review of the experiments until 1989 has been given previously by Crooker [17]; we present only the salient facts here:

. As shown in Figure 7.4, BPIII is the highest temperature blue phase,

appearing either between BPII and the isotropic phase, or, at higher chiralities, between BPI and the isotropic phase. Like BPI, but unlike BPII, it is characterized by a temperature range which increases monotonically with increasing chirality. . The visual appearance of BPIII is foggy, quite unlike the structured appearance of BPI or BPII [14]. Since it appears only at higher chiralities, it is often bluish or grayish in color. At very high chiralities, it may be invisibleÐindistinguishable from the isotropic phaseÐto the naked eye. Visually, therefore, BPIII appears to be closer in structure to the isotropic phase than to BPI or BPII. . Like BPI and BPII, BPIII selectively re¯ects circularly polarized light [52], [53]. Unlike the cubic blue phases, however, the spectrum is quite broad (@100 nm) [132]. Also, while BPI and BPII exhibit several Bragg peaks (corresponding to various crystal planes), BPIII exhibits only one peak. . BPIII exhibits rotatory power, the magnitude of which, in general, decreases with temperature and chirality [133], [134]. At the BPI/BPII±BPIII transition, and at the BPIII±isotropic transition, the rotatory power jumps discontinuously.

7. Blue Phases

213

. As can be seen from Figure 7.3, there is a small heat capacity peak between BPIII and the cubic blue phases and a much larger peak between BPIII and the isotropic phase. From the heat capacity data, BPIII therefore appears to be closer in structure to BPII than to the isotropic phase, in contrast to the visual appearance. . As shown in Figure 7.12, measurements of the shear elasticity and the visosity of BPIII show higher values than for the neighboring BPII and isotropic phases [25]. . Electron microscopy of freeze-fractured samples of BPIII have shown a ®lamentary structure, with details on the order of a tenth of the pitch in the helical phase [135]. The conclusion from this preliminary evidence was that BPIII was a separate phase, thermodynamically distinct from the cubic and isotropic phases, but amorphous and chiral. More recent optical and thermodynamic measurements have proved revealing. Based on a suggestion by Koistinen and Keyes [136] that BPIII and the isotropic phase have the same symmetry, one would expect the BPIII±isotropic coexistence line (see Figure 7.4) to end in a critical point at some higher, unexplored chirality in the temperature± chirality plane. In this respect, the situation is analogous to the liquid±gas coexistence line, which, since liquids and gases have the same symmetry, also ends at a liquid±gas critical point. Using light scattering [137], [138] and rotatory power techniques [139], Collings and coworkers were able to show that the discontinuities present at low-chirality BPIII±isotropic transitions disappeared at very high chiralities. Figure 7.18 shows the rotatory power data in several chiral±racemic mixtures of the highly chiral liquid crystal CE2 (British Drug House). In these measurements, the chirality is adjusted by mixing nL moles of the left-handed enantiomer of CE2 with nR moles of right-handed CE2. The chirality is then proportional to the ``chiral mole fraction'' X ˆ …nL ÿ nR †=…nL ‡ nR †. Note

Figure 7.18. Rotatory power F=d versus temperature T near the BPIII± isotropic transition in three mixtures of varying chirality. The material is a mixture of left- and right-handed enantiomers of CE2 (British Drug House), with chiral mole fraction X ˆ 0:35; 0:45; 1. These values bracket the critical point near Xc ˆ 0:45 (from Kutjnak et al. [139]).

214

P.P. Crooker

Figure 7.19. Nonadiabatic (circles) and adiabatic (smooth curve) calorimetry data for two chiralities of CE2. Peaks appearing only in the nonadiabatic data indicate a ®rst-order transition. The BPIII±isotropic peak evident for X ˆ 0:40 has vanished for X ˆ 0:45, which is thought to be near the critical point at Xc . The region denoted by ‡ signs indicates phase-shift anomalies in the adiabatic data, which also accompany a ®rst-order transition (from Kutjnak et al. [139]).

that the step in the rotatory power at low chiralities, which is the signature of a ®rst-order transition, disappears at higher chirality. More evidence for a critical point comes from adiabatic and nonadiabatic scanning calorimetry measurements [138], [139], a comparison of which yields the latent heat e¨ects associated with a ®rst-order phase transition. Figure 7.19 shows data from both techniques for two chiralities of a CE2 sample. The sharp peaks, which are due to the nonabiatic technique only, represent ®rst-order helical±BPI and BPI±BPIII transitions; the BPIII± isotropic peak only shows ®rst-order behavior for X ˆ 0:40. Repeated runs for various chiralities establishes that, for this system, Xc A 0:45. A complete experimental analysis of the rotatory power and calorimetric experiments has been performed by Kutnjak et al. [139], who conclude that the data is consistent with mean ®eld behavior. Since that time, however, new theory has appeared which allows di¨erent conclusions.

7.6.2

Theory

As experiments developed a description of BPIII behavior, a number of early models were presented. Up to 1989, these models included the following:

7. Blue Phases

215

. The pretransitional ¯uctuation model assumed that BPIII is simply a mani-

.

.

.

.

festation of pretransitional ¯uctuations in the isotropic phase at the blue phase±isotropic boundary [3], [4]. This idea was discounted [56] by the fact that the observed BPIII scattering [53] is several orders of magnitude too large for pretransitional ¯uctuations. In any case, the calorimetry data [26] rule out the pretransitional ¯uctuation model. The emulsion model [103] suggested that BPIII is just an emulsion of helical droplets in an isotropic background. This model requires pure materials to have a two-phase region, in violation of thermodynamics, and is also inconsistent with present experiments. The double-twist model of Hornreich et al. [140] suggests that BPIII is a spaghetti-like tangle of double-twist cylinders (see Figure 7.6), but with a scalar order parameter which dies away in Gaussian-like fashion with increasing distance from the cylinder axis. This model has not been completely discounted to date, and is not in disagreement with present data. The cubic domain model includes the possibility that BPIII retains local cubic structure, but only over short correlated regions. For these regions, Collings [134] has suggested a BPII-like sc O 2 structure, whereas Belyakov et al. [84] have proposed a BPI-like bcc O 8 structure. Experiments by Yang et al. [141] have shown, however, that BPIII exhibits only a single broad selective re¯ection peak without any indication of the higher orders expected from a cubic structure. Also, Kitzerow et al. [142] have shown that, for negative dielectric anisotropy, BPIII may be aligned by an electric ®eld, which sharpens and greatly intensi®es (by tenfold) the selective re¯ection peak. Measurements of the line shape and intensity of this peak versus electric ®eld indicate that the coupling between the order parameter Q and the electric ®eld E is proportional to Q 2 E n with n being equal to 2. The cubic model requires n ˆ 4 [143], which rules it out; but the double-twist model, for which n ˆ 2, is still allowed. In fact, a locally cubic model becomes the double-twist model in the limit of small correlation length, so perhaps the di¨erence between the two is merely a question of the size of the correlation length. The icosahedral model [144], [145], [146], [147] has been the most analytically tractable so far. Starting from the Landau theory of Hornreich and Shtrikman [32], a structure with reciprocal lattice vectors derived from the vertices of a regular icosahedron is assumed. Using the 12 vertex vectors and the 30 edge vectors of the icosahedron, a structure with a free energy considerably higher than that of, say, the bcc O 5 cubic phase is found. Such a high-energy structure would, of course, be unstable to O 5 . If, however, the symmetry is broken by shifting the phases of the edge vectors, the energy can be lowered until the structure is almost, but not quite, stable. Nevertheless, the theory may have overlooked some feature, such as higher-order terms, which would make the structure stable. E¨orts to verify this model

216

P.P. Crooker

were attempted by Crooker and coworkers, who searched for, but were unable to ®nd, icosahedral selective re¯ections [141]. In addition, the electric ®eld experiments mentioned earlier, which ruled out cubic BPIII structures, also ruled out icosahedral structures, for which one expects n V 4. At present, therefore, due to the lack of experimental and theoretical evidence for the icosahedral structure, attention has turned to other explanations for BPIII. The notion [136], and accumulating experimental evidence [139], that the the BPIII±isotropic coexistence line might terminate in a critical point has stimulated recent theoretical work by Lubensky and Stark [148]. Letting Q ˆ Qij denote the customary order parameter tensor, they assume a new, pseudoscalar order parameter formed from the chiral term in the free energy hci ˆ hQ
…`  Q†i ˆ heijk Qil `j Qkl i:

…7:20†

In a conventional helical structure, hci @ S 2 q, where S ˆ h…3 cos 2 y ÿ 1†=2i is the scalar order parameter and q ˆ 2p=P; in general, hci contains information on both the chirality and the order itself. This new order parameter is discontinuous across the coexistence line, with the discontinuity decreasing to zero at the critical point. In this sense, the BPIII±isotropic transition is analogous to the liquid±gas transition. Using c and Q, a coarse-grained Landau±Ginzburg±Wilson Hamiltonian is written which admits the mixing of the physical variables temperature and chirality to produce the theoretical scaling ®elds. The critical behavior of this model is then shown to be in the universality class of the three-dimensional Ising model. The authors are also able to derive expressions near the critical point for the light-scattering correlation functions and the rotatory power and claim qualitative agreement with the earlier optical experiments of Collings and coworkers [137], [138]. More recently, Anisimov et al. [149] have further exploited the idea of mixing physical variables by quantitatively comparing two equations of stateÐa mean ®eld model and a scaling modelÐwith the heat capacity and optical data of Kutnjak et al. [139]. Writing the scaling ®elds incorporated in the models as linear combinations of temperature and chirality, they ®nd the mixing for the BPIII±isotropic transition to be more extreme than the analogous liquid±gas transition. Comparing the ®ts for their two equations of state, the authors claim that the scaling equation of state gives the better ®t [see Figure 7.20], which is evidence that this transition does, after all, belong to the same university class as the three-dimensional Ising model. Finally, we recall the discussion of phase diagrams in Section 7.3.4. In the bond-orientational order model of Trebin et al. [45], [46], only limited success was achieved by modeling BPIII as the O 5 cubic phase with bondorientational order. Also, in the same authors' ¯uctuation-dominated model [47], a strong indication of BPIII was impossible to achieve.

7. Blue Phases

217

Figure 7.20. Molar heat capacity data near the BPIII± isotropic critical point X ˆ 0:40. Dots are data; solid lines are ®ts to the mean-®eld and scaling theories (from Anisimov et al. [149]).

7.7

Conclusions

Blue phases now seem to be fairly well understood, and a generic picture of the temperature±chirality phase diagram for blue phases has emerged, as shown in Figure 7.21. The survival of BPI and BPIII at high chiralities is well established, as is the existence of a critical point terminating the BPIII± isotropic coexistence line. BPII occupies a temperature region between BPI and BPIII, but only over a limited chirality range. The symmetry of BPI appears to be bcc O 8 …I 41 32†; that of BPII sc O 2 …P42 32†. Although the short-range structure of BPIII has not yet been strictly determined, it has the same overall symmetry as the isotropic phase, a fact which allows for the appearance of the BPIII±isotropic critical point. One speculation is that BPIII consists of small correlated regions of double twist. Theoretically, production of a phase diagram which brings out all the features of the experimental phase diagram has been a daunting problem and is still ongoing. Application of small electric ®elds results in electrostriction of the BPI and BPII lattices. Except for the case of anomalous dispersion in BPI, this behavior has been explained theoretically. Larger ®elds cause hexagonal and tetragonal phases to be stabilized. The hexagonal phase has received theoretical justi®cation; the tetragonal phase still awaits theoretical treatment. Finally, we opened this chapter by noting the many surprising ways in which matter can aggregate. Blue phases are only one defect phase to

218

P.P. Crooker

Figure 7.21. Generic phase diagram showing temperature T versus chirality q for blue phases. Universal features include the survival of BPI and BPIII, but not BPII, at high chirality and termination of the BPIII±isotropic transition at critical point c.p.

appear in chiral liquid crystals, the other being the twist grain boundary (TGB) phase [150], [151]. Both phases are a result of chirality. Recently Pansu and coworkers [152], [153] have investigated a liquid crystal where the blue phases are adjacent to the TGB phase (no intermediate helical phase), and the blue phase, as revealed by X-rays, has smectic order. This work is very preliminary, but once again nature has presented us with a new, unanticipated chiral structure to try and understand. Thus, research will continue on chiral materials in general and blue phases in particular. As scientists, we can only hope that the end is nowhere near in sight. Acknowledgments. The author would like to thank the Kent State Physics Department and Liquid Crystal Institute for their hospitality while part of this manuscript was being written. Thanks also to P. Collings and C. Vause for useful discussions on BPIII, and to D. Beck and E. Dodson for a critical reading of the manuscript.

References [1] P.G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed., Oxford University Press, London, 1993, Sec. 6.5. [2] P.G. de Gennes, Mol. Cryst. Liq. Cryst. 12, 193 (1971). [3] S.A. Brazovskii and S.G. Dmitriev, Zh. Eksp. Teor. Fiz. 69, 979 (1975); [Sov. Phys. JETP 42, 497 (1975)].

7. Blue Phases

219

[4] S.A. Brazovskii and V.M. Filev, Zh. Eksp. Teor. Fiz. 75, 1140 (1978); [Sov. Phys. JETP 48, 573 (1978)]. [5] J.P. Sethna, Phys. Rev. Lett. 51, 2198 (1983). [6] J.P. Sethna, D.C. Wright, and N.D. Mermin, Phys. Rev. Lett. 51, 467 (1983). [7] S.A. Brazovskii, Zh. Eksp. Teor. Fiz. 68, 175 (1975); [Sov. Phys. JETP 41, 85 (1975)]. [8] P.P. Crooker, Mol. Cryst. Liq. Cryst. 98, 31 (1983). [9] J.P. Sethna, in: Theory and Applications of Liquid Crystals (edited by J.L. Ericksen and D. Kinderlehrer), Springer-Verlag, New York, 1987, p. 305. [10] E. Dubois-Violette and B. Pansu, Mol. Cryst. Liq. Cryst. 165, 151 (1988). [11] W.F. Brinkman and P.E. Cladis, Physics Today 35, 48 (1982). [12] S. Meiboom, J.P. Sethna, P.W. Anderson, and W.F. Brinkman, Phys. Rev. Lett. 46, 1216 (1981). [13] See R.M.J. Cotterill, in: Ordering in Strongly Fluctuation Condensed Matter Systems (edited by T. Riste), Plenum, New York, 1980 and references therein. [14] H. Stegemeyer and K. Bergmann, in: Liquid Crystals of One- and Two-Dimensional Order (edited by W. Helfrich and G. Heppke), Springer-Verlag, Berlin, 1980, p. 161. [15] H. Stegemeyer, Th. Blumel, K. Hiltrop, H. Onusseit, and F. Porsch, Liq. Cryst. 1, 1 (1986). [16] V.A. Belyakov and V.E. Dmitrienko, Uspekhi Fiz. Nauk 146, 369 (1985); [Sov. Phys. Uspekhi 28, 535 (1985)]. [17] P.P. Crooker, Liq. Cryst. 5, 751 (1989). [18] P.E. Cladis, in: Theory and Applications of Liquid Crystals (edited by J.L. Ericksen and D. Kinderlehrer), Springer-Verlag, New York, 1987, p. 73. [19] T. Seideman, Rep. Prog. Phys. 53, 659 (1990). [20] D.C. Wright and N.D. Mermin, Rev. Mod. Phys. 61, 385 (1989). [21] A. Armitage and F.P. Price, J. Chem. Phys. 66, 3414 (1977). [22] D. Armitage and F.P. Price, J. Phys. 36, C1±133 (1975). [23] A. Armitage and R.J. Cox, Mol. Cryst. Liq. Cryst. Lett. 64, 41 (1980). [24] K. Bermann and H. Stegemeyer, Z. Naturforsch. 34a, 251 (1979). [25] R.N. Kleiman, D.J. Bishop, R. Pindak, and P. Taborek, Phys. Rev. Lett. 53, 2137 (1984). [26] J. Thoen, Phys. Rev. A 37, 1754 (1987). [27] G. Voets, H. Martin, and W. van Dael, Liq. Cryst. 5, 871 (1989). [28] K. Tanimoto, P.P. Crooker, and G.C. Koch, Phys. Rev. A 32, 1893 (1985). [29] J.D. Miller, P.R. Battle, P.J. Collings, D.K. Yang, and P.P. Crooker, Phys. Rev. A 35, 3959 (1987). [30] M.B. Atkinson and P.J. Collings, Mol. Cryst. Liq. Cryst. 136, 141 (1986). [31] D.K. Yang and P.P. Crooker, Phys. Rev. A 35, 4419 (1987). [32] R.M. Hornreich and S. Shtrikman, J. Phys. 41, 335 (1980); 42, 367(E) (1981). [33] R.M. Hornreich and S. Shtrikman, in: Liquid Crystals of One- and TwoDimensional Order (edited by W. Helfrich and G. Heppke), Springer-Verlag, Berlin, 1980, p. 185. [34] R.M. Hornreich and S. Shtrikman, Phys. Rev. A 24, 635 (1981). [35] R.M. Hornreich and S. Shtrikman, Phys. Lett. 84A, 20 (1981). [36] H. Grebel, R.M. Hornreich, and S. Shtrikman, Phys. Rev. A 28, 1114 (1983). [37] H. Grebel, R.M. Hornreich, and S. Shtrikman, Phys. Rev. A 30, 3264 (1984). [38] H. Kleinert and K. Maki, Fortschr. Phys. 29, 219 (1981).

220 [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79]

P.P. Crooker R.M. Hornreich and S. Shtrikman, Mol. Cryst. Liq. Cryst. 165, 183 (1988). L. Longa, D. Monselesan, and H.-R. Trebin, Liq. Cryst. 5, 889 (1989). J.P. Sethna, Phys. Rev. B 31, 6278 (1985). S. Meiboom, M. Sammon, and D. Berreman, Phys. Rev. A 28, 3553 (1983). S. Meiboom, M. Sammon, and W.F. Brinkman, Phys. Rev. A 27, 438 (1983). B. Pansu, J. Phys. II France 5, 573 (1995). L. Longa and H.-R. Trebin, Phys. Rev. Lett. 71, 2757 (1993). L. Longa and H.-R. Trebin, Liq. Cryst. 21, 243 (1996). J. Englert, L. Longa, H. Stark, and H.-R. Trebin, Phys. Rev. Lett. 81, 1457 (1998). K. Bergmann and H. Stegemeyer, Ber. Bunsenges. Phys. Chem. 82, 1309 (1978). K. Bergmann, P. Pollmann, G. Scherer, and H. Stegemeyer, Z. Naturforsch. 34a, 253 (1979). K. Bergmann and H. Stegemeyer, Z. Naturforsch. 34a, 1031 (1979). H. Stegemeyer, Phys. Lett. 79A, 425 (1980). S. Meiboom and M. Sammon, Phys. Rev. Lett. 44, 882 (1980). S. Meiboom and M. Sammon, Phys. Rev. A 24, 468 (1981). D.L. Johnson, J.H. Flack, and P.P. Crooker, Phys. Rev. Lett. 45, 641 (1980). R.M. Hornreich and S. Shtrikman, Phys. Lett. 82A, 345 (1981). R.M. Hornreich and S. Shtrikman, Phys. Rev. A 28, 1791 (1983); 28, 3669(E) (1983). V.A. Belyakov, V.E. Dmitrenko, and S.M. Osadchii, Zh. Eksp. Teor. Fiz. 83, 585 (1982); [Sov. Phys. JETP 56, 322 (1982)]. K. Tanimoto and P.P. Crooker, Phys. Rev. A 29, 1566 (1984). J.W. Gorman Jr. and P.P. Crooker, Phys. Rev. A 31, 910 (1985). J.L. Birman, R.M. Hornreich, and S. Shtrikman, Phys. Rev. A 39, 1590 (1989). J.H. Flack, P.P. Crooker, and R.C. Svoboda, Phys. Rev. A 26, 723 (1982). J.W. Gorman Jr. and P.P. Crooker, Phys. Rev. A 31, 910 (1985). P. Pieranski, E. Dubois-Violette, F. Rohten, and L. Strelecki, J. Phys. 42, 53 (1981). P.E. Cladis, T. Garel, and P. Pieranski, Phys. Rev. Lett. 57, 2841 (1986). P. Pieranski and P.E. Cladis, Phys. Rev. A 35, 355 (1987). B. JeÂroÃme, P. Pieranski, V. Godec, G. Haran, and C. Germain, J. Phys. France 49, 837 (1988). R.J. Miller and H.F. Gleeson, Phys. Rev. E 52, 5011 (1995). R.J. Miller and H.F. Gleeson, J. Phys. II France 6, 909 (1996). R.J. Miller, H.F. Gleeson, and J.E. Lydon, Phys. Rev. Lett. 77, 857 (1996). M. Marcus, J. Phys. 42, 61 (1981). M. Marcus, Phys. Rev. A 25, 2272 (1982). M.J. Costello, S. Meiboom, and M. Sammon, Phys. Rev. A 29, 2957 (1984). D.W. Berreman, S. Meiboom, J.A. Zasadzinski, Phys. Rev. Lett. 57, 1737 (1986). H. Onusseit and H. Stegemeyer, Z. Naturforsch. 36a, 1083 (1981). H. Onusseit and H. Stegemeyer, J. Cryst. Growth 61, 409 (1983). H. Onusseit and H. Stegemeyer, Z. Naturforsch. 38a, 1114 (1983). Th. BluÈmel and H. Stegemeyer, J. Cryst. Growth 66, 163 (1984). Th. BluÈmel and H. Stegemeyer, Z. Naturforsch. 40a, 260 (1985). R. Barbet-Massin, P.E. Cladis, and P. Pieranski, Phys. Rev. A 30, 1161 (1984).

7. Blue Phases

221

[80] P. Pieranski, R. Barbet-Massin, and P.E. Cladis, Phys. Rev. A 31, 3912 (1985). [81] P. Pieranski, P.E. Cladis, and R. Barbet-Massin, J. Phys. 47, 129 (1986). [82] P. Pieranski, P.E. Cladis, T. Garel, and R. Barbet-Massin, J. Phys. (Paris) 47, 139 (1986). [83] T.K. Brog and P.J. Collings, Mol. Cryst. Liq. Cryst. 60, 65 (1980). [84] V.A. Belyakov, E.I. Demikhov, V.E. Dmitrienko, and V.K. Dolganov, Zh. Eksp. Teor. Fiz. 89, 2035 (1985); [Sov. Phys. JETP 62, 1173 (1986)]. [85] P.J. Collings, Mol. Cryst. Liq. Cryst. 113, 277 (1984). [86] M.B. Atkinson and P.J. Collings, Mol. Cryst. Liq. Cryst. 136, 141 (1986). [87] J. Ennis, J.E. Wyse, and P.J. Collings, Liq. Cryst. 5, 861 (1989). [88] D. Bensimon, E. Domany, and S. Shtrikman, Phys. Rev. A 28, 427 (1983). [89] T. Yamada and E. Fukada, Jap. J. Appl. Phys. 12, 68 (1973). [90] P.H. Keyes and D.B. Ajgaonkar, Phys. Lett. 64A, 298 (1977). [91] H. Stegemeyer and P. Pollmann, Mol. Cryst. Liq. Cryst. 82, 123 (1982). [92] T. Harada and P.P. Crooker, Phys. Rev. Lett. 34, 1259 (1975). [93] D.S. Mahler, P.H. Keyes, and W.B. Daniels, Phys. Rev. Lett. 36, 491 (1976). [94] P.H. Keyes and C.C. Yang, J. Phys. Colloq. 40, C3±376 (1979). [95] P.P. Crooker and W.G. Laidlaw, Phys. Rev. A 21, 2174 (1980). [96] N. Clark, S.T. Vohra, and M.A. Handschy, Phys. Rev. Lett. 52, 57 (1984). [97] P.E. Cladis, P. Pieranski, and M. Joanicot, Phys. Rev. Lett. 52, 542 (1984). [98] A.I. Feldman, P.P. Crooker, and L.M. Goh, Phys. Rev. A 35, 842 (1987). [99] W. Kuczynski and H. Stegemeyer, Naturwiss. 67, 310 (1980). [100] Th. BluÈmel and H. Stegemeyer, Liq. Cryst. 3, 195 (1988). [101] H.-S. Kitzerow, Mol. Cryst. Liq. Cryst. 207, 981 (1991). [102] D. Lubin and R.M. Hornreich, Phys. Rev. A 36, 849 (1987). [103] P.L. Finn and P.E. Cladis, Mol. Cryst. Liq. Cryst. 84, 159 (1982). [104] F. Porsch, H. Stegemeyer, and K. Hiltrop, Z. Naturforsch. 39a, 475 (1984). [105] H. Stegemeyer and F. Porsch, Phys. Rev. A 30, 3369 (1984). [106] F. Porsch and H. Stegemeyer, Chem. Phys. Lett. 125, 319 (1986). [107] H. Stegemeyer and B. Spier, Chem. Phys. Lett. 133, 176 (1987). [108] G. Heppke, M. Krumrey, and F. Oestreicher, Mol. Cryst. Liq. Cryst. 99, 99 (1983). [109] G. Heppke, H.-S. Kitzerow, and M. Krumrey, Mol. Cryst. Liq. Cryst. 150b, 265 (1987). [110] P. Pieranski, P.E. Cladis, and R. Barbet-Massin, Liq. Cryst. 5, 829 (1989). [111] R.M. Hornreich, M. Kugler, and S. Shtrikman, Phys. Rev. Lett. 54, 2099 (1985). [112] R.M. Hornreich, M. Kugler, and S. Shtrikman, J. Phys. Colloque C3, C3±47 (1985). [113] F. Porsch and H. Stegemeyer, Liq. Cryst. 2, 395 (1987). [114] N.R. Chen and J.T. Ho, Phys. Rev. A 35, 4886 (1987). [115] P. Pieranski, P.E. Cladis, and R. Barbet-Massin, J. Phys. Lett. 46, 973 (1985). [116] M. Jorand and P. Pieranski, J. Phys. (Paris) 48, 1197 (1987). [117] G. Heppke, B. JeÂroÃme, H.-S. Kitzerow, and P. Pieranski, Liq. Cryst. 5, 813 (1989). [118] R.M. Hornreich and S. Shtrikman, Liq. Cryst. 5, 777 (1989). [119] R.M. Hornreich and S. Shtrikman, Phys. Rev. A 41, 1978 (1990). [120] P. Pieranski and P.E. Cladis, Phys. Rev. A 35, 355 (1987). [121] V.E. Dmitrienko, Liq. Cryst. 5, 847 (1989).

222

P.P. Crooker

[122] H. Stark and H.-R. Trebin, Phys Rev. A 44, 2752 (1991). [123] L. Longa, M. Zelazna, H.-R. Trebin, and J. Moscicki, Phys. Rev. E 53, 6067 (1996). [124] M. Zelazna, L. Longa, H.-R. Trebin, and H. Stark, Phys. Rev. E 57, 6711 (1998). [125] G. Heppke, H.-S. Kitzerow, and M. Krumrey, Mol. Cryst. Liq. Cryst. Lett. 1, 117 (1985). [126] G. Heppke, B. JeÂroÃme, H.-S. Kitzerow, and P. Pieranski, J. Phys. France 50, 549 (1989). [127] G. Heppke, B. JeÂroÃme, H.-S. Kitzerow, and P. Pieranski, J. Phys. France 50, 2991 (1989). [128] H.F. Gleeson and H.J. Coles, Liq. Cryst. 5, 917 (1989). [129] H.-S. Kitzerow, P.P. Crooker, S.L. Kwok, and G. Heppke, J. Phys. France 51, 1303 (1990). [130] H.-S. Kitzerow, P.P. Crooker, S.L. Kwok, J. Xu, and G. Heppke, Phys. Rev. A 6, 3442 (1990). [131] H.-S. Kitzerow, P.P. Crooker, J. Rand, J. Xu, and G. Heppke, J. Phys. II France 2, 179 (1992). [132] E.l. Demikhov, V.K. Dolganov, and S.P. Krylova, Pis'ma Zh. Eksp. Teor. Fiz. 42, 15 (1985); [JETP Lett. 42, 15 (1985)]. [133] E.I Demikhov and V.K. Dolganov, Sov. Phys. JETP Lett. 38, 445 (1983). [134] P.J. Collings, Phys. Rev. A 30, 1990 (1984). [135] J.A.N. Zasadzinski, S. Meiboom, M.J. Sammon, and D.W. Berreman, Phys. Rev. Lett. 57, 364 (1986). [136] E.P. Koistenen and P.H. Keyes, Phys. Rev. Lett. 74, 4460 (1995). [137] J.B. Becker and P.J. Collings, Mol. Cryst. Liq. Cryst. 265, 163 (1995). [138] Z. Kutnjak, C.W. Garland, J.L. Passmore, and P.J. Collings, Phys. Rev. Lett. 74, 4859 (1995). [139] Z. Kutnjak, C.W. Garland, C.G. Schatz, P.J. Collings, C.J. Booth, and J.W. Goodby, Phys. Rev. E 53, 4955 (1996). [140] R.M. Hornreich, M. Kugler, and S. Shtrikman, Phys. Rev. Lett. 48, 1404 (1982). [141] D.K. Yang, P.P. Crooker, and K. Tanimoto, Phys. Rev. Lett. 61, 2685 (1988). [142] H.-S. Kitzerow, P.P. Crooker, and G. Heppke, Phys. Rev. Lett. 67, 2151 (1991). [143] R.M. Hornreich, Phys. Rev. Lett. 67, 2155 (1991). [144] R.M. Hornreich and S. Shtrikman, Phys. Rev. Lett. 56, 1723 (1986). [145] R.M. Hornreich and S. Shtrikman, Phys. Lett. A 115, 451 (1986). [146] D.S. Rokhsar and J.P. Sethna, Phys. Rev. Lett. 56, 1727 (1986). [147] V.M. Filev, Pis'ma Zh. Eksp. Teor. Fiz. 43, 523 (1986); [JETP Lett. 43, 677 (1986)]. [148] T.C. Lubensky and H. Stark, Phys. Rev. E 53, 714 (1996). [149] M.A. Anisimov, V.A. Agayan, and P.J. Collings, Phys. Rev. E 57, 582 (1998). [150] S.R. Renn and T.C. Lubensky, Phys. Rev. A 38, 2132 (1988). [151] J.W. Goodby, M.A. Waugh, S.M. Stein, E. Chin, R. Pindak, and J.S. Patel, Nature 337, 449 (1989). [152] B. Pansu, M.H. Li, and H.T. Nguyen, J. Phys. II France 7, 751 (1997). [153] B. Pansu, M.H. Li, and H.T. Nguyen, European Phys. J. B 2, 143 (1998).

8

Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect Christian Bahr

8.1

Introduction

Due to special symmetry conditions, the presence of chiral molecules in tilted smectic phases leads to the appearance of a spontaneous electric polarization within each smectic layer [1], [2]. Magnitude and direction of the polarization are coupled to the magnitude and direction of the molecular tilt giving rise to unique properties of tilted as well as orthogonal smectic phases of chiral molecules in external electric ®elds. In this chapter, the basic properties of smectic-C and smectic-A phases of chiral low-molecular weight molecules are summarized, mainly from the viewpoint of physics. Section 8.2 gives a short introduction to the basic structure of smectic phases, especially smectic-A and smectic-C, since this is the ®rst chapter of the volume dealing with smectic phases. The next section is concerned with the ferroelectric properties of the smectic-C phase: the origin of the spontaneous polarization, the polarization-tilt angle coupling, dielectric properties, and electro-optic applications are brie¯y described. The coupling between molecular tilt and electric polarization, which leads to the spontaneous polarization in tilted smectic phases, is present also in orthogonal smectic phases where the induction of an electric polarization results in the induction of a tilt angle; Section 8.4 gives an overview of this electroclinic e¨ect in the smectic-A phase of chiral molecules. The main focus of Section 8.5 is then on the relation between the molecular chirality and the ferroelectric and electroclinic properties of smectic-C and smectic-A phases. The last Section 8.6 gives a brief introduction to a very recent topic: the ferroelectric properties observed in nonchiral molecules possessing a bent core. Since there are already many well-written and comprehensive reviews on ferroelectric liquid crystals [3], this chapter has a more introductory or tutorial character. It is also far beyond the scope of this chapter to give a complete account of the literature published in this ®eld, which has grown enormously in the last two decades; I hope that the works cited in [3] will serve the reader as a comprehensive source to the literature.

223

224

C. Bahr

8.2

Smectic Phases

Just as in the nematic phase, the rod-like molecules in smectic liquid±crystal phases are orientationally ordered: the molecules tend to align their long axis along the director ~ n. In addition to this nematic-like orientational order, all smectic phases show a positional order of their molecules along at least one spatial dimension, describable either as a density wave with the periodicity of about one molecular length or as an arrangement of the molecules in layers. There can be di¨erent orientations between the density wave vector or layer normal ~ z and the director ~ n: in ``orthogonal'' smectic phases ~ n and ~ z are parallel, whereas in ``tilted'' smectic phases ~ n is inclined by a tilt angle y with respect to ~ z. There are several kinds of positional order within the layers; one may distinguish between three groups: Phases possessing only liquid-like short-range order within their layers: smectic-A (orthogonal) and C (tilted). Phases possessing a so-called bond orientational order within their layers: smectic-Bhex (orthogonal), I (tilted), and F (tilted). These phases are also designated as hexatic smectic phases. The molecules are locally ordered in hexagons and the orientation of these hexagons (i.e., the orientation of a ®ctive ``bond'' between neighboring molecules) is long-range ordered. In a three-dimensional sample, the bond orientational order is correlated across the layers of the sample; thus, it is a true long-range three-dimensional orientational order. The di¨erence between smectic-I and F consists of the direction of the tilt with respect to the local hexagons. Phases possessing a true long-range three-dimensional positional order: (smectic-)Bcryst , E (both orthogonal), J; G; K, and H (all four tilted). The molecules are positionally ordered within the layers and there are correlations between the layers, i.e., there is a true long-range positional order of the molecules. These phases were initially designated as ``smectic'' but are now generally considered as crystalline phases (which still possess an orientational disorder concerning the rotation of the molecules around their long and short axes). A comprehensive description can be found in the books by Pershan [4] and Gray and Goodby [5]. This chapter will be mainly concerned with smectic-A and C phases, the structures of which are considered in the following.

8.2.1

Structure of Smectic-A and Smectic-C Phases

Smectic-A (orthogonal) and smectic-C (tilted) are the least-ordered smectic phases. The layer structure of these phases actually consists of a onedimensional density wave. The shape of the density wave appears to be almost purely sinusoidal as can be concluded from the absence or very weak intensity of higher-order re¯ections in X-ray experiments. Thus, one should

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

225

Figure 8.1. Schematic drawings of the molecular arrangements in the smectic-A (right) and smectic-C (left) phase; ~ z: layer normal, ~ n: director, y: tilt angle.

have in mind that the layer structure of smectic-A and C phases is usually less well de®ned1 than shown in Figure 8.1 which gives a schematic sketch of the structures of these phases. A detailed analysis of the X-ray re¯ection peak of bulk smectic-A phases shows that the peak shape is described by a power law singularity rather than by the d-function of a true Bragg peak [6]. This is in accordance with theoretical considerations predicting the absence of true long-range order in systems which are ordered along only one spatial dimension. The positional order along the smectic layer normal is thus of a quasi-long-range order type which means that the corresponding correlation function shows an algebraic decay. This di¨erence between true and quasi-long-range order is, however, of little relevance with respect to the ferroelectric and electroclinic properties of these phases. Quite often both phases, smectic-A and C, appear in the same compound, smectic-C then being the low-temperature phase to smectic-A. There are, however, many compounds possessing a smectic-A phase without underlying smectic-C phase; also compounds showing a direct transition from smectic-C to nematic or even isotropic are known. The smectic-A±smectic-C phase transition is in the vast majority of compounds a second-order transition, a few substances showing ®rst-order transitions are known [7], [8]. The transition can be described by a complex order parameter C: C ˆ y exp…if†;

…8:1†

where y is the magnitude of the tilt angle between layer normal and director and f speci®es the direction of the tilt within the smectic layer plane. Typical values of y far below the transition to the smectic-A phase are between 30 and 40 . Approaching the transition from below, y either decreases continuously (in the second-order case) to zero or it jumps (in the ®rst-order case) to zero 1 The situation might be di¨erent at interfaces, e.g., at the surfaces of freely suspended ®lms where the positional order is considerably enhanced compared to the bulk.

226

C. Bahr

typically from a value between 10 and 20 . The two-component order parameter places the smectic-A±smectic-C transition into the same universality class as the super¯uid transition in liquid helium and since de Gennes noted this analogy in 1973 [9], many studies were concerned with the critical behavior of the transition. Most studies conclude that the transition is of the mean-®eld type with an unobservably small critical region [10], [11] but true critical behavior is also reported [12]. The schematic sketch in Figure 8.1 shows the ``usual'' smectic-C phase in which the tilt direction is locally constant. There are phases in which the tilt direction alternates by G180 from layer to layer or shows even more complicated behaviors. These phases will be discussed in the following chapter on antiferroelectric liquid crystals. Essentially, the structural features described above apply to both nonchiral and chiral compounds. However, the presence of chiral molecules in smectic-A and C phases results in additional properties and structures not present in phases of nonchiral substances. These are the ferroelectric properties and the electroclinic e¨ect, which will be discussed in detail in Sections 8.3 and 8.4, and the helical structure in the smectic-C phase.

8.2.2

Helical Structure in the Chiral Smectic-C Phase

If a smectic-C phase is formed by chiral moleculesÐregardless of whether a chiral compound exhibits a smectic-C phase by itself or a smectic-C phase of nonchiral molecules is doped with a chiral additiveÐa helical structure appears which is in some aspects similar to the helical structure in a cholesteric liquid crystal. The helical structure of the chiral smectic-C phase had been recognized in the early 1970s [13], [14], [15], well before the ferroelectric properties of this phase were realized. In the chiral smectic-C phase, the helix is formed by a precession of the director ~ n around the layer normal ~ z, the azimuthal tilt direction f changes by a small amount when going from layer to layer as is shown schematically in Figure 8.2. The helical pitch p can be in the range from the wavelength of visible light up to arbitrary large values, in most compounds p amounts to a few mm. Within the major part of the smectic-C phase range, p is only weakly temperature-dependent: it increases slightly with increasing temperature. If the smectic-C phase transforms into a smectic-A phase at higher temperatures, p reaches a maximum of a few tenths of a degree below the smectic-C±smectic-A transition temperature TAC and then drops to a signi®cantly smaller but ®nite value at TAC (see Figure 8.3) [16], [17]. Although both the helical structure and the ferroelectric properties of the chiral smectic-C phase have their origin in the molecular chirality, they are not directly linked to each other. In suitable binary mixtures, one can practically unwind the helix without loosing the ferroelectricity, and it is also possible to prepare mixtures with practically zero ferroelectric polarization possessing a helical structure. On the other hand, there is some relation be-

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

227

Figure 8.2. Helical structure of the chiral smectic-C phase. The position of the director ~ n on the tilt cone changes by a small amount from layer to layer. One full helical pitch corresponds usually to several hundred smectic layers.

tween helix and electric polarization: unlike as in the cholesteric phase, where the helical structure corresponds to a spontaneous twist of the ~ n ®eld, the helical structure of the chiral smectic-C phase corresponds to a combination of a spontaneous twist and bend. In a bent ~ n ®eld, suitable molecules can produce a ¯exoelectric polarization [18]. The steep temperature dependence of the helical pitch p near TAC can be traced to a nonlinear relation between electric polarization and tilt angle (which will be discussed in the next section) which in¯uences the magnitude of p via the ¯exoelectric coupling.

Figure 8.3. Typical temperature-dependence of the helical pitch p in the chiral smectic-C phase (compound DOBAMBC, from [17]).

228

C. Bahr

8.3

The Smectic-C Phase of Chiral Molecules: Ferroelectric Properties

As is described in the following, the presence of chiral molecules leads to the ~s in each layer of the appearance of a spontaneous electric polarization P ~s points ~s is coupled to the tilt direction, P smectic-C phase. The direction of P along either ~ z ~ n or ÿ…~ z ~ n†. Because of the helical structure in the smectic~s spirals with the periodicity of the helical pitch C phase of chiral molecules, P p around the layer normal ~ z and in a macroscopic sample the spontaneous polarization averages to zero. The designation ``ferroelectric'' is thus in a strict sense not correct (``helielectric'' has been proposed instead); nevertheless, the term ``ferroelectric'' has become common usage for the smectic-C phase of chiral molecules and is retained in this chapter. Experiments on freely suspended ®lms have proven the existence of a spontaneous polarization in ®lms consisting of only two or three smectic layers [19], [20], the designation ``ferroelectric'' is thus justi®ed at least for a single smectic-C layer.

8.3.1

Symmetry of the Chiral Smectic-C Phase and Molecular Origin of the Spontaneous Polarization

In all smectic phases known to date, the directions ~ n and ÿ~ n are equivalent. In each smectic layer, the same number of molecules point toward the ``top'' or the ``bottom'' establishing complete up±down symmetry even if the molecules are strongly asymmetric like 4,4 0 -alkylcyanobiphenyls.2 Accordingly, there is always at least one twofold symmetry axis in a direction perpendicular to ~ n. In the smectic-C phase, this symmetry axis is perpendicular to the tilt plane which is de®ned as the plane containing ~ z and ~ n (in the sketch of the smectic-C phase in Figure 8.1, the twofold axis is perpendicular to the plane of the paper). If the smectic-C phase is built up by nonchiral molecules, the tilt plane is also a mirror plane. Thus, the local symmetry of the nonchiral smectic-C phase is described by the point group3 C2 h . If the constituent molecules are chiral, the mirror plane is removed and the symmetry is reduced to the point group C2 . The twofold symmetry axis remains as the only symmetry element and becomes a polar axis. This means that any molecular physical property, which is represented by a vector, is not ``automatically'' averaged to zero by the symmetry of the structure but can remain, to a certain extent, as a macroscopic property of the phase. A necessary condition is 2 Such strongly polar molecules often form a more complex smectic layer structure such as double layers or partial double layers but the up±down symmetry is always retained. 3 The description of liquid±crystal phases by symmetry point groups applies to the time average of the structure.

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

229

that this vector property possessesÐin the time-averaged structureÐa nonzero component parallel to the direction of the polar axis. Let us consider in the following the molecular dipole moment and the time-averaged structure in the smectic-C phase. Liquid±crystal molecules usually possess a permanent dipole moment ~ m oriented at some arbitrary angle with respect to the long molecular axis. Because the smectic-C phase is a liquid, the molecules change their position and rotate around their long and short axes. The rotation around the short axis is strongly biased, positions parallel to the director ~ n are strongly preferred compared to positions perpendicular to ~ n; nevertheless, the rotational motion around the short axis preserves the up±down symmetry in the smectic layers. Concerning the rotation of the molecules around their long axis, there is no preferred azimuthal position as long as the molecule is in a nematic or smectic-A phase. In the smectic-C phase, the tilt of ~ n with respect to the layer normal ~ z breaks the axial symmetry around the long molecular axis. A single chiral molecule then experiences a rotational potential which leadsÐon time averageÐto the preference of a certain azimuthal position compared to all other azimuthal orientations. Thus, there remains a net dipole moment h~ mi corresponding to the azimuthal orientation of the molecule in this preferred position (the magnitude of h~ mi, however, may be rather small compared to ~ m, since one can expect only a small distortion of the axial symmetry around ~ n). Now we have still to consider that there are equal numbers of ``up''- and ``down''-oriented molecules in a smectic layer. The time-averaged dipole h~ mi of the ``up''-molecules consists of a component h~ mi? perpendicular to the tilt plane and a component h~ mik parallel to the tilt plane, the corresponding mik , i.e., the comcomponents of the ``down''-molecules are h~ mi? and ÿh~ ponents of h~ mi which are parallel to the tilt plane cancel but the perpendicmi? add up to a ular components h~ mi? remain. All these components h~ ~s of the smectic-C layer, macroscopic spontaneous electric polarization P directed perpendicular to the tilt plane and thus parallel to the polar C2 -axis. It is useful to realize the di¨erences between chiral and nonchiral molecules. The molecular tilt breaks the axial symmetry around ~ n for both chiral and nonchiral molecules but there are di¨erences concerning the shape of the resulting rotational potential. In a microscopic model [21], the azimuthal potential V of the rotation around the long axis can be described for a single chiral molecule as a sum of two terms describing polar and quadrupolar ordering V …j† ˆ a1 y cos…j ‡ d1 † ‡ a2 y 2 cos‰2…j ‡ d2 †Š:

…8:2†

Here j gives the azimuthal position of the molecule, the phases d1; 2 determine the preferred positions, and the coe½cients a1; 2 describe the contribution of each term to the total potential. For a nonchiral molecule, the polar cos j part vanishes and only the cos 2j term remains. For a chiral molecule, there is just one absolute minimum of V …j† in the range between 0 and 2p, but a nonchiral molecule shows two equivalent preferred azimuthal ori-

230

C. Bahr

Figure 8.4. Molecular structure of the DOBAMBC.

entations which di¨er by p. Thus, concerning the net dipole moment h~ mi resulting from these potentials, a single nonchiral molecule shows only a nonzero component h~ mik parallel to the tilt plane which is, in the ensemble of all molecules, averaged to zero by the up±down symmetry of the smectic layer. With the symmetry considerations described above in mind, Meyer and coworkers [1] prepared speci®cally a chiral compound possessing a smecticC phase, the now well-known compound DOBAMBC (see Figure 8.4). DOBAMBC served for the ®rst experimental proofs of a spontaneous ~s in liquid crystals, as was demonstrated by a linear electric polarization P response to an applied ®eld in electro-optic experiments [1] and by the appearance of a shear-induced electric polarization in the absence of an applied ~s in DOBAMBC amounts to A3 nC/cm 2 which ®eld [22]. The magnitude of P corresponds to A0:01 Debye/molecule. These numbers show that only a small portion of the molecular dipole moment ~ m is left in the net molecular dipole h~ mi? of the time-averaged structure. In other words, the molecular rotation around the long axis is probably not substantially hindered or slowed down in the smectic-C phase; dielectric measurements have shown that there is no change of the corresponding relaxation frequency at the smectic-A±smectic-C transition [23]. On the other hand, optical four-wave mixing experiments yielded results indicating a slowing down of the reorientation time about the long molecular axis [24], [25]. Furthermore, there are compounds showing values of Ps two orders of magnitude larger than DOBAMBC [26], [27], [28]. In these compounds approximately one-third of the molecular dipole contributes to Ps ; thus, the detailed picture of the molecular rotation in the ferroelectric smectic-C phase has still to be clari®ed.

8.3.2

Relation Between Tilt Direction and Polarization Direction: The Sign of the Spontaneous Polarization

~s results from the comAs described above, the spontaneous polarization P ponents h~ mi? of the net molecular dipole which are directed perpendicular to the plane containing layer normal ~ z and director ~ n. Accordingly, there are ~s , it for a given compound in principle two possibilities for the direction of P could point along either ~ z ~ n or ÿ…~ z ~ n†. These two cases (usually only one is realized in a given compound) are distinguished by assigning a ``sign'' to ~s form a right-handed system and a negative ~s , namely a positive sign, if ~ z~ nP P sign for the opposite case [29] (see Figure 8.5). Optical antipodes always

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

231

Figure 8.5. Relation between layer normal ~ z, director ~ n, and spontane~s for ous polarization P compounds with positive ~s . or negative sign of P

~s and in a racemic mixture the magnitude of Ps is possess opposite signs of P zero. ~s resulting from a certain molecular structure? Can one predict the sign of P ~s depends, in the end, on the direction of the net molecular The sign of P dipole h~ mi of the time-averaged structure. A corresponding prediction is, besides from the empirical rules [30], [31], to some extent possible using the Boulder model [32], according to which the time-averaged structure of a molecule in the smectic-C phase has to ®t into a zig-zag-shaped binding site. The alkyl chains of the molecule are then less tilted than the core; this geometry is justi®ed by a steric model [33] for the origin of the smectic-C phase ~s is determined from the preand by X-ray measurements [34]. The sign of P ferred molecular conformation by weighing the di¨erent conformations ®tting to the binding site with their corresponding steric strain energies. ~s sign In the vast majority of compounds, a given enantiomer possesses a P which does not change throughout the whole temperature range of the smectic-C phase. There are, however, a few substances which show a spon~s at a certain temperature within the smectic-C taneous sign inversion of P phase [35], [36]. Such a behavior can occur when the net dipole h~ mi of a molecule in its time-averaged structure is directed almost parallel to the tilt plane. The remaining component h~ mi? perpendicular to the tilt plane, which is not cancelled by the molecular up±down symmetry, is then rather small ~s is always small in these compounds. When and indeed the magnitude of P the temperature is varied, even small changes of the direction of h~ mi may be su½cient to rotate h~ mi to the other side of the tilt plane, thus resulting in an ~s . inversion of its component h~ mi? and a sign inversion of P

8.3.3

Relation Between Tilt Magnitude and Polarization Magnitude: Bilinear and Biquadratic Coupling Between Tilt Angle and Spontaneous Polarization

According to the symmetry considerations described above, the magnitude ~s should be proportional to the magnitude of the spontaneous polarization P ~s j increases of the tilt angle y [1]. Experimentally, one usually observes that jP monotonically with decreasing temperature similar to y but the detailed

232

C. Bahr Figure 8.6. Temperature-dependence of tilt angle y, spontaneous polarization Ps , and ratio Ps =y in the ferroelectric smectic-C phase of the compound DOBAMBC (from [37]).

behavior is more complicated than a simple proportionality: for most com~s j=y shows immediately below the smectic-A±smecticpounds, the ratio jP C transition a pronounced increase and then remains weakly temperaturedependent in the major part of the smectic-C phase [37] (see Figure 8.6). The probable explanation of this behavior is that the rotational potential, which is experienced by a molecule in the smectic-C phase, consists of a combination of polar and quadrupolar ordering terms as given in (8.2). The polar ordering term alone, which exists only for chiral compounds, would result in ~s and y but the presence of the quadrupolar a pure linear relation between P term (the temperature-dependence of which di¨ers from that of the polar term), which exists for chiral as well as nonchiral compounds, results in a ~s and y [21]. nonlinear relation between P ~s and y is On a more phenomenological level, the relation between P described by Landau theories. Since the predictions of Landau models will be used in several of the following sections, a short introduction is given here. The basic part gy of the Landau free energy g consists, for chiral and nonchiral compounds, of a series of powers of the tilt order parameter gy ˆ 12 a…T ÿ Tc †y 2 ‡ 14 by 4 ‡ 16 cy 6 :

…8:3†

For a second-order transition, a, b, and c are positive constants and Tc is the phase transition temperature. The y 6 term is necessary for a correct description of the temperature-dependence of the tilt angle and heat capacity [10]. If a spatially modulated structure is to be described, it is convenient to use a n two-component tilt vector ~ x ˆ …xx ; xy †, corresponding to the projection of ~ onto the smectic layer plane, instead of y: gy ˆ 12 a…T ÿ Tc †…xx2 ‡ xy2 † ‡ 14 b…xx2 ‡ xy2 † 2 ‡ 16 c…xx2 ‡ xy2 † 3 :

…8:4†

Additional terms are then introduced to establish a coupling between tilt y and electric polarization P: a bilinear term, gPl , corresponding to the polar term in (8.2), and a biquadratic term, gPq , corresponding to the quadrupolar term in (8.2) [38], [39]:

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

233

gPl ˆ ÿCPy ‡ 12 eÿ1 P 2 ;

…8:5†

gPq ˆ ÿ 12 WP 2 y 2 ‡ 14 hP 4 :

…8:6†

In the spatially modulated case, (8.5) and (8.6) have to be modi®ed using the ~ ˆ …Px ; Py † which is perpendicular to the tilt vector ~ x. polarization vector P The constants C and W describe the strength of the bilinear and biquadratic coupling; the P 2 and P 4 terms are necessary to stabilize the system, e has the physical meaning of an electric susceptibility, it can be written as e ˆ w0 e0 with e0 being the vacuum permittivity and w0 (electric susceptibilities and dielectric constants are used in this chapter as dimensionless or relative quantities) corresponds to the electric susceptibility of the uncoupled system …C ˆ W ˆ 0†. Writing the Landau free energy g as g ˆ gy ‡ gPl results in a simple proportionality between P and y: …8:7† P ˆ Cw0 e0 y: The addition of the biquadratic coupling, g ˆ gy ‡ gPl ‡ gPq , leads to a nonlinear relation between P and y (which cannot be expressed as a simple equation), and is su½cient to obtain the experimentally observed (see Figure 8.6) nonlinear relation between P and y. The nonlinear P…y† dependence close to the smectic-C±smectic-A transition may also be responsible for the strong variation of the helical smectic-C pitch near the transition [17]. The helical structure is described by gq : "    #   qxy qxy 2 qxx qxx 2 1 ÿ xy ‡ ‡ 2K 3 gq ˆ ÿL xx : …8:8† qz qz qz qz The ®rst term in (8.8) with the Lifshitz coe½cient L favors the helical modulation of ~ x and the second term with the elastic constant K3 describes the corresponding elastic energy; z is the coordinate along the layer normal. The helical director ®eld is accompanied by a ¯exoelectric polarization which is described by gPf containing the ¯exoelectric coupling constant m:   qxy qxx ‡ Py : …8:9† gPf ˆ ÿm Px qz qz Taking only the bilinear P±y coupling into account, i.e., g ˆ gy ‡ gPl ‡ gq ‡ gPf , one obtains for the helical pitch p (resp. the corresponding wave vector q): 2p L ‡ w0 e0 mC ˆ ; …8:10† qˆ p K3 ÿ w0 e0 m 2 which is a constant value (independent of temperature). But including the biquadratic P±y coupling into g leads, as a consequence of the nonlinear P…y† relation which e¨ects p via the ¯exoelectric coupling, to a temperaturedependent p…T† resembling the experimentally observed behavior. A detailed description of the Landau model, including all g terms given above, can be found in [40].

234

C. Bahr

As described above, there is some correspondence between the bilinear/ biquadratic P±y couplings of the phenomenological Landau model and the polar/quadrupolar rotational potentials of the microscopic model. Indeed, the most pronounced deviation from a proportionality between Ps and y, the ~s , can be obtained in both models, either by assuming sign inversion of P certain values of the Landau parameters [41] or, in the molecular rotation description, by a slight modi®cation of the rotational potential (8.2) [42].

8.3.4

Dielectric Properties: Goldstone Mode and Soft Mode

Like the usual dielectrics, the ferroelectric smectic-C phase possesses contributions to its dielectric permittivity which are based on the deformation of molecular electron shells and the orientation of permanent molecular dipoles. The dielectric properties at low frequencies, however, are dominated by additional contributions, the Goldstone mode and the soft mode [43], ~s and the which result from the presence of the spontaneous polarization P coupling between Ps and y. ~s spirals around the In an undistorted bulk sample, the direction of P smectic layer normal ~ z because of the helical variation of the tilt direction and the net polarization is zero. The Goldstone mode is the dielectric mode ~s caused by an electric which results from changes to the local direction of P ~ ~ ®eld E. A small electric ®eld E perpendicular to ~ z causes a torque on the local ~s vectors of the smectic layers. The helical structure is slightly distorted and P ~ appears. The relative dielectric strength a nonzero net polarization along E DeG of the Goldstone mode can be in the range of several hundreds, and its relaxation frequency nG is usually of the order of 1 kHz or less [44], [45]. Besides the magnitude of Ps , the helical structure of the smectic-C phase determines the properties of the Goldstone mode: within the framework of the Landau model one expects DeG z p 2 =K3 and nG z K3 = p 2 ( p being the helical pitch and K3 the bend elastic constant) [46]. Whereas the Goldstone mode is associated with changes in the direction of tilt and polarization, the soft mode arises from changes in the magnitude of ~s is these quantities. Consider a smectic-C layer in which the direction of P ~ already along the direction of an external ®eld E. An increase of the tilt ~ direction. The inmagnitude y can then increase the polarization in the E crease of y is balanced by a restoring force which is expected to vanish at the second-order transition to the smectic-A phase. Thus, the dielectric strength Des and the relaxation frequency ns of the soft mode should become large (resp. small) near the smectic-C±smectic-A transition. Experimentally, however, the soft mode in the smectic-C phase is usually covered by the stronger Goldstone mode, but it can be studied on the smectic-A side of the phase transition where the Goldstone mode does not exist. A more detailed description of the soft mode will thus be given in Section 8.4 on the properties of the smectic-A phase of chiral molecules.

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

235

Figure 8.7. Idealized con®guration of the SSFLC cell. Depending on the polarity of ~ the director ~ the applied dc ®eld E, n switches between two di¨erent positions on the tilt cone, corresponding to two di¨erent positions of the optical axis in a plane parallel to the cell plates. In real devices the smectic layer planes are not perfectly perpendicular to the cell plates but show a chevron structure with a kink inside the cell.

8.3.5

Electro-Optic E¨ects

~ is applied to the chiral smectic-C phase in a diIf a large dc electric ®eld E ~s rection perpendicular to the smectic layer normal, the polarization vector P ~ of each layer is forced to align along E and the tilt direction is in each layer ~ There is no longer a helical structure, the director ~ perpendicular to E. n is ~ If aligned throughout the sample parallel to the plane perpendicular to E. the polarity of the applied ®eld is inverted, polarization and tilt direction are also inverted, the position of ~ n is then still within the plane perpendicular to ~ but at an angle relative to its previous position corresponding to twice the E tilt angle y between ~ n and ~ z (see Figure 8.7). Neglecting the weak biaxiality of the smectic-C phase, one could think of a uniaxial system whose optical axis can be switched between two positions, di¨ering by an angle 2y, by inverting the polarity of an applied dc ®eld. The polar director switching described above is one of the two basic features of the surface-stabilized ferroelectric liquid±crystal (SSFLC) cell which was ®rst described in 1980 [47]. The second important feature is that the unwinding of the helical director structure is achieved by surface e¨ects rather than by a large ®eld strength. The idealized structure of an SSFLC cell is the so-called bookshelf geometry in which the smectic layer planes are strictly perpendicular to the two glass plates of the cell. The plate surfaces are treated so that the molecules at the surfaces are forced to align parallel to the surface; in the smectic-C phase ~ n is constrained at the surface to two ~s pointing either toward or away from the surface. If the cell positions with P is thin enough (U2 mm in practice), the structure inside the cell is dominated by the surface alignment and a helix cannot develop. The electric ®eld is then

236

C. Bahr

~s and switch the director, it does not need to renecessary only to inverse P main to keep the helix unwound. Thus, the switching between the two stable ~ n positions can be made in a bistable way by short dc pulses of opposite polarity. Between crossed polarizers, the intensity of transmitted light varies n and a polarizer axis), i.e., maximum as sin 2 …2f† (f being the angle between ~ contrast is achieved for a switching angle of 45 corresponding to y ˆ 22:5 . The main advantage over nematic displays consists of the short switching ~s , the times which can be in the ms range, depending on the magnitude of P strength of the applied ®eld, and the substance viscosity. The SSFLC cell described above with its perfect bookshelf geometry and bistable switching corresponds, however, to an ideal state which is usually not achieved by real SSFLC devices. A bookshelf geometry of the smectic layers can be prepared in the smectic-A phase but on cooling to the smecticC phase a chevron-like layer structure develops [48], i.e., the smectic layer planes make some angle d > 0 with the cell normal and possess a kink of 180 ÿ 2d somewhere inside the cell. A consequence is the appearance of socalled zig-zag walls which separate domains possessing di¨erent director con®gurations [49]. Furthermore, there is some relaxation between structures with and without an applied ®eld. Despite these di½culties, the SSFLC e¨ect is the basis of several applications which are now at the edge of commercialization like display devices [50], spatial light modulators [51], and light shutters using a total internal re¯ection geometry [52] or waveguides [53]. There are also electro-optic e¨ects using either a di¨erent geometry of surface stabilization or a completely di¨erent mechanism: In the twisted ferroelectric smectic-C cell [54] the molecules form in the zero ®eld state a quarter helix which is removed when a dc ®eld of either polarity is applied; the optical e¨ect is achieved in the same way as in a twisted nematic cell. Compounds with a short chiral smectic-C pitch in a thick cell are used for the distorted helix ferroelectric (DHF) device [55]: this e¨ect uses the optical di¨erence between the zero-®eld state characterized by a fully developed short-pitch helix, and structures with a distorted or almost unwound helix in the presence of an applied ®eld; optically addressed spatial light modulators can take advantage of the DHF e¨ect [56]. Further applications of ferroelectric liquid crystals are switchable di¨raction gratings [57].

8.4 8.4.1

The Smectic-A Phase of Chiral Molecules: Electroclinic E¨ect Origin and Basic Properties of the Electroclinic E¨ect

In the smectic-A phase there is usually a complete rotational symmetry around the long molecular axis. As described in Section 8.3.1, the tilt of the molecules in the smectic-C phase breaks this symmetry and leads to a polar

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

237

ordering and the appearance of a spontaneous polarization. Another way to break the cylindrical symmetry around the director ~ n in the smectic-A phase, ~ perpendicular to ~ is the application of a dc electric ®eld E n. The presence of ~ causes a bias of the molecular rotation around the long axis and leads to E ~E , based on a small part of the permanent an induced electric polarization P ~ In the case of nonchiral transverse molecular dipole moment, along E. ~ the ~ molecules, the magnitude of PE is determined only by the strength of E, magnitude of the molecular dipole, and the rotational motion around the long axis. But if the molecules are chiral, there is a mechanism by which an ~E can be increased by a tilt of ~E is produced: P additional contribution to P ~ Indeed, such a ®eld-induced the molecules in a direction perpendicular to E. tilt in the smectic-A phase of chiral molecules was already observed [58], [59] experimentally along with the ®rst studies of the ferroelectric smectic-C phase. The magnitude of the ®eld-induced tilt angle yE in the smectic-A phase is proportional to E, provided the system is not close to the transition to the smectic-C phase. Typical values of the ratio yE =E, which depends on temperature as described in the next section, are in the range of a few degrees per kV/cm (see Figure 8.8). Reversal of the ®eld direction leads to a reversal of the induced tilt direction and the response time of the electroclinic e¨ect is usually less than 1 ms [60], [61]. These properties are promising features for an electro-optic e¨ect using the same geometry as for the SSFLC cell. In contrast to the SSFLC e¨ect, the director cannot only be switched between two positions corresponding to an applied ®eld of ‡E or ÿE, rather every angular position in between is adjustable; thus, a linear electro-optic modulator can be realized. A problem, which can partly be overcome by using a combination of multiple cells [62], is the low modulation depth resulting

Figure 8.8. Electric ®eldinduced tilt angle in the smectic-A phase of the chiral compound A7 (the molecular structure is shown at the top). The voltage is applied across a 10 mm thick cell. The transition to the smecticC phase is at 73:4  C (from [61]).

238

C. Bahr

from the small values of yE . A value of yE ˆ 22:5 , corresponding to maximum optical modulation, can be reached near the transition to the smectic-C phase but then the e¨ect becomes nonlinear in E and there is a slowing down of the response time.

8.4.2

Soft Mode Behavior near the Transition to the Smectic-C Phase

The induction of a tilt angle yE in the smectic-A phase of chiral molecules ~E in yields an energetic gain resulting from the increase in the polarization P ~ the presence of the ®eld E. This is balanced by the elastic energy necessary to rotate the director ~ n from its zero ®eld position. The corresponding elastic constant should vanish when the second-order transition to the smectic-C phase is approached. The expected relation between y and E near the transition can be described by the Landau model. Neglecting the helical structure in the smectic-C phase and the biquadratic P±y coupling, we can write the Landau free energy g ˆ gy ‡ gPl ; then, from (8.3) and (8.5) we get for g in the presence of a ®eld E (see Section 8.3.3 for the meaning of the quantities): g ˆ 12 a…T ÿ Tc †y 2 ‡ 14 by 4 ‡ 16 cy 6 ÿ CPy ‡

1 P 2 ÿ EP: 2w0 e0

…8:11†

Minimization according to qg=qy ˆ 0 leads to P ˆ Cx0 e0 y ‡ w0 e0 E;

…8:12†

from which a function E ˆ E…T; y† can be obtained Eˆ

1 ‰‰a…T ÿ Tc † ÿ C 2 w0 e0 Šy ‡ by 3 ‡ cy 5 Š: Cw0 e0

It is useful to de®ne an electroclinic tilt susceptibility wy :   qy ; wy ˆ qE E!0

…8:13†

…8:14†

which is obtained from (8.13) in the limit E ! 0 as wy ˆ

Cw0 e0 : a…T ÿ Tc †

…8:15†

The electroclinic tilt susceptibility wy should be accompanied by an additional dielectric contribution wPy which increases the total electric susceptibility of the chiral enantiomer compared to its racemate wPy ˆ

C 2 w02 e0 : a…T ÿ Tc †

…8:16†

Thus, wy and wPy should diverge when the transition to the smectic-C phase is approached, an experimental example is shown in Figure 8.9. Although the

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

239

tilt susceptibility, as de®ned by (8.14), may become very large near Tc , the values of the ®eld-induced tilt angle behave of course di¨erently: Close to Tc the y versus E curves start with a large slope, but for larger E, according to the nonlinear y±E relation of (8.13) and as observed experimentally [63], [64], the slope decreases. The linearity desired for the electro-optic application is obtained only su½ciently far from Tc . On the other hand, measurements of y as a function of ®eld and temperature near the smectic-A±smectic-C provide a convenient possibility for the determination of the various Landau coe½cients [65], [66], [67]. The divergence of wy is accompanied by a slowing down of the dynamical response. The induced tilt angle modulation shows a Debye-like relaxation, the frequency of which decreases linearly as Tc is approached [61]. We should note that (8.15) is only an approximation. Because of the helical structure in the chiral smectic-C phase, the divergence of wy is in principle incomplete since a real divergence would be obtained only as a response to a helicoidal electric ®eld [68], [69]. Including the helical structure into the Landau model leads to a modi®cation of (8.15) and a truncation of the divergence with a ®nite value of wy at Tc , similar to the case of a ®rst-order transition. However, whereas the truncation at a ®rst-order smectic-A± smectic-C transition can be observed experimentally, measurements of wy around a tricritical point, where the transition changes from ®rst-order to second-order, have shown that the in¯uence of the helix on the divergence of wy is probably beyond experimental resolution [70].

8.4.3

Induced Tilt Angle and Induced Polarization

The electroclinic e¨ect is a result of the coupling between tilt and polarization. The polarization PE , which is induced by an external ®eld E in the smectic-A phase of chiral molecules, consists of a part P0 which is present in every dielectric (orientation and electronic polarization), and a part Py which is due to the P±y coupling and should show a similar behavior as the induced tilt angle y. While it is di½cult to separate Py and P0 exactly, measurements of the total polarization in the smectic-A phase and around the smectic-A±smectic-C transition indicate at least qualitatively that Py and y show very similar behavior [65]. The separation between Py and P0 is easier for the corresponding electric susceptibilities wPy and w0 . The value of w0 can be measured if the racemate of a given compound is available, wPy then corresponds to the di¨erence w ÿ w0 (w being the total electric susceptibility of the chiral enantiomer); if the temperature range of the smectic-A phase is not too large, w0 is found to show a constant value [61]. In the chiral enantiomer, the electric susceptibility w and the tilt susceptibility wy can be measured simultaneously as a function of temperature. It is found that the optically measured values of wy behave exactly as the di¨erence w ÿ w0 , both show a divergence-like increase as the transition to the smectic-C phase is approached (see Figure 8.9). The

240

C. Bahr

Figure 8.9. Simultaneous measurement of electric susceptibility (relative dielectric constants of the chiral enantiomer and racemate, top) and tilt susceptibility wy (labeled as y=E, middle) and ratio k ˆ …w ÿ w0 †e0 =wy (bottom) in the smectic-A phase of the chiral compound A7 (from [61]).

ratio …w ÿ w0 †e0 =wy , which corresponds to the Ps =y ratio of the smectic-C phase, is constant throughout the smectic-A phase [61], [71]. The apparent absence of a biquadratic P±y coupling in the smectic-A phase may be due to the fact that the susceptibility measurements were made with small ®elds and thus at very small induced tilt angles, where the quadrupolar ordering term

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

241

in the rotational potential (8.2) becomes negligible. Indeed, recent measurements of tilt angle and total polarization on both sides of the smectic-A± smectic-C transition have indicated that e¨ects of the biquadratic P±y coupling are seen only on the smectic-C side at fairly large tilt angles [72].

8.5

Relation Between Ferroelectric and Electroclinic Properties and Molecular Chirality: Chiral±Racemic Studies

By mixing a chiral liquid±crystal compound with its optical antipode, systems possessing arbitrary values of the enantiomeric excess can be designed. If a chiral compound shows smectic-C and smectic-A phases, the racemate, i.e., the 1:1 mixture of the two antipodes, also exhibits these phases but the ferroelectric properties of the smectic-C phase and the electroclinic e¨ect in the smectic-A phase are lost. This o¨ers the unique possibility to study a given system with and without ferroelectricity or with a variable markedness of its ferroelectric properties.

8.5.1

The Smectic-A±Smectic-C Transition

In many solid-state ferroelectrics the transition to the ferroelectric phase is driven by electrostatic interactions resulting in the appearance of the spon~s , i.e., the polarization is the primary order parameter taneous polarization P of the phase transition. In ferroelectric liquid crystals, the transition to the ferroelectric smectic-C phase in chiral compounds is driven by the same tiltproducing interactions as in nonchiral compounds, the primary order parameter of the transition being in both cases the molecular tilt whereas the polarization is a secondary order parameter. The basically linear coupling between tilt and polarization leads nevertheless to some analogies in the behavior of ferroelectric liquid crystals and solid-state ferroelectrics, like a diverging electric susceptibility or a shift of the transition temperature in an electric ®eld [73]. If the linear tilt±polarization coupling is very strong, the di¨erence between primary and secondary order parameters becomes blurred, the transition to the ferroelectric smectic-C phase in chiral compounds is then additionally driven by the electric polarization. In such compounds, the smectic-C±smectic-A transition temperature should be higher in the chiral enantiomer than in the corresponding racemate. This e¨ect has been quantitatively demonstrated for compounds possessing large Ps values in their ferroelectric smectic-C phase, the transition temperature di¨erence DTAC between chiral enantiomer and racemate amounts to values between 0.8 K and 3.2 K [74], [75], [76], [77], [78]. As shown in Figure 8.10, DTAC varies with the square of the mole fraction of the enantiomeric excess. This square dependence is in accordance with

242

C. Bahr Figure 8.10. Smectic-A± smectic-C transition temperature as a function of the mole fraction xee of the enantiomeric excess for the compound A7; xee ˆ G1 corresponds to the pure optical antipodes, xee ˆ 0 to the racemate (from [76]).

predictions of the simplest Landau description of the smectic-A±ferroelectric smectic-C transition. The transition from smectic-A to smectic-C occurs when the minimum in the Landau free energy g…y† shifts from y ˆ 0 to y > 0. For a system described by g ˆ gy : gy ˆ 12 a…T ÿ Tc †y 2 ‡ 14 by 4 ‡ 16 cy 6 ;

…8:17†

the transition occurs either at T ˆ Tc (second-order transition, a; b; c > 0) or at T ˆ Tc ‡ 3b 2 =16ac (®rst-order transition, a; c > 0 and b < 0). Adding the bilinear tilt±polarization coupling term gPl [see (8.5)] to g results in a shift DTAC which is in both cases, ®rst- and second-order, given by DTAC ˆ

C 2 w 0 e0 : a

…8:18†

The tilt±polarization coupling constant C equals zero in the racemate and some ®nite value GC  in the pure chiral enantiomers. The experimentally observed square dependence of the transition temperature on the enantiomeric excess is then obtained by assuming a simple linear relation C ˆ C  xee . Are there additional e¨ects, beyond the shift of the transition temperature, on the smectic-A±smectic-C phase transition in chiral±racemic systems? The transition is, in the vast majority of compounds, of the second-order type, the ®rst examples for a ®rst-order smectic-A±smectic-C transition were found in chiral compounds possessing large Ps values [7], [8]. It has been even observed, that a weakly ®rst-order transition in the chiral enantiomer becomes continuous in the racemate [79]. However, it seems that chirality and/or large spontaneous polarization are not the primary reasons for the occurrence of ®rst-order smectic-A±smectic-C transitions, since ®rst-order transitions were also found in racemic or nonchiral compounds [80], [81] (the above-mentioned second-order transition in a racemate results probably from an increased width of the smectic-A temperature range in the racemate compared to the chiral enantiomer). One relevant factor for a ®rst-order

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

243

Figure 8.11. Spontaneous polarization Ps as a function of the mole fraction xee of the enantiomeric excess for the compound A7 at TAC ÿ T ˆ 1:5 K (from [74]).

smectic-A±smectic-C transition may consist of the magnitude of the transverse dipole moment [80], the ®rst-order nature of the transition and the large Ps value would then result from a common molecular origin but would not depend on each other.

8.5.2

The Ferroelectric Properties in the Smectic-C Phase

Optical antipodes possess the same magnitude of the spontaneous polariza~s with ~s in their smectic-C phase but show opposite directions of P tion P respect to the molecular tilt direction; in the racemate Ps equals zero. One thus expects a simple linear relation between Ps and the enantiomeric excess xee . Experimentally, the behavior of the spontaneous polarization has been studied in [74], [75], [78], [82]. In [74], [78] a strictly linear Ps …xee † dependence at constant temperature di¨erence to the smectic-C±smectic-A transition temperature TAC was observed (see Figure 8.11). The linear Ps …xee † dependence is in accordance with the simple Landau description which uses the Landau free energy g ˆ gy ‡ gPl [(8.3) and (8.5)] and predicts Ps z y [(8.7)]. Proportionality between the tilt±polarization coupling constant C and xee then results in the experimentally observed proportionality Ps z xee ; provided the shift of the transition temperature TAC with varying xee is taken into account. Slight deviations from the above-described proportionality between Ps and xee were also observed [75], [82]. These deviations have been attributed either to the in¯uence of the biquadratic tilt±polarization coupling [82], or, on a more molecular level, to small di¨erences in the intermolecular interactions experienced by a molecule in a racemic environment compared to a chiral environment [75]. ~s varies essentially linearly In chiral±racemic mixtures, the magnitude of P with the enantiomeric excess. Important for applications is the more general case of a mixture of a nonchiral smectic-C host with a chiral dopant. The ~s and the concentration of the chiral dopant in such mixrelation between P

244

C. Bahr

tures can range from a simple proportionality to di¨erent types of nonlinear ~s can occur at a certain composition [83], behaviors; even a sign inversion of P [84], [85]. Another quantity which has been studied in chiral±racemic systems [75], and which is important for applications, is the rotational viscosity gf corresponding to the motion of the director ~ n on the tilt cone at constant tilt magnitude y. On ®rst approximation, the switching time t in response to the reversal of a ®eld E is given as gf ; …8:19† tˆ Ps E which implies that switching may be very fast if Ps is large enough. However, it was observed in chiral±racemic mixtures that gf shows a pronounced linear increase with xee and thus with the magnitude of Ps , i.e., Ps and gf may be increased by the same molecular interactions. Thus, it is di½cult to design ferroelectric mixtures suitable for applications simply by maximizing the magnitude of Ps .

8.5.3

The Electroclinic E¨ect in the Smectic-A Phase

The ferroelectric liquid±crystal compounds which have been studied in chiral±racemic systems possess large values of the spontaneous polarization Ps , i.e., these compounds show a strong bilinear coupling between tilt angle and polarization. The behavior in chiral±racemic systems of these compounds can be well described assuming a simple proportionality of the bilinear P±y coupling constant C and the enantiomeric excess xee . This applies also for the electroclinic e¨ect in the smectic-A phase which has been studied in [74], [77]. Figure 8.12 shows the electroclinic tilt susceptibility wy as a function of xee at constant temperature di¨erence to the transition to the smectic-C phase. The observed proportionality between wy and xee is well in

Figure 8.12. Electroclinic tilt susceptibility wy (labeled as tilt angle/®eld strength) as a function of the mole fraction xee of the enantiomeric excess for the compound A7 at T ÿ TAC ˆ 1:0 K (from [74]).

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

245

agreement with (8.15) if C z xee . Also the dielectric soft mode contribution wPy depends on the enantiomeric excess as expected from C z xee and (8.16): wPy varies with the square of xee [74], [77]. In contrast to the ferroelectric switching time in the smectic-C phase, the dynamics of the electroclinic e¨ect does not depend on the chirality. The soft mode relaxation frequency was observed to show the same linear dependence on …T ÿ TAC † in several mixtures possessing di¨erent values of the enantiomeric excess [77].

8.6

Ferroelectricity in Liquid±Crystal Phases of Nonchiral Bent-Core Molecules

The ferroelectric properties of the chiral smectic-C phase and the electroclinic e¨ect of the smectic-A phase appeared as a result of the symmetry breaking caused by the presence of chiral molecules. One can think of smectic phases in which nonchiral molecules arrange themselves in a polar order [86], and it seems that such phases were recently observed, indeed experimentally. The molecules which establish these phases are not chiral but possess a bent core resembling a bow- or banana-like shape [87]; a second class of nonchiral liquid crystals showing polar ordering consists of certain polymer±monomer mixtures [88]. Experimental study of the bent-core compounds is still at a very early stage, their ability to produce a spontaneous electric polarization had ®rst been recognized in 1996 [89]. Optical studies of freely suspended ®lms have led to a structural model [90] for one of the occurring phases (labeled as smectic-CP) which is brie¯y described in the following. In the layers of the smectic-CP phase, the molecular bows point along a common direction ~ b thereby establishing a polar axis parallel to the layer plane (see Figure 8.13). In addition, the molecules are tilted in a direction

Figure 8.13. Schematic sketch of the local molecular arrangement in a single layer of the smectic-CP phase; shown isÐin two side viewsÐonly one out of two possible chiral arrangements of layer normal ~ z, tilt direction ~ x, and bow direction or polar axis ~ b.

246

C. Bahr

perpendicular to ~ b. Thus, layer normal ~ z, tilt direction ~ x, and polar axis ~ b can arrange in two ways possessing opposite handedness or chirality. One can now think of di¨erent arrangements or alternation patterns of tilt and polarization direction as one goes from layer to layer, e.g., the direction of ~ x could be the same in all layers while the direction of ~ b alternates by G180 from layer to layer. Such a structure would be antiferroelectric and racemic (because the chiral handedness alternates from layer to layer). A second possibility is the alternation of both, ~ x and ~ b; this structure is also antiferroelectric but chiral. These two antiferroelectric structures may be considered as possible ground states which can be transformed to ferroelectric structures by the presence of a large enough dc electric ®eld. The above-described considerations have been con®rmed by the di¨erent behaviors of ®lms consisting of either an even or an odd number of smectic layers, and by electro-optic switching studies which show the spontaneous formation of macroscopic chiral domains possessing opposite handedness. An important point is that the inversion of the spontaneous polarization, induced by the inversion of an applied ®eld, occurs preferably by the inversion of the tilt direction ~ x and not by an inversion of the chiral handedness (inversion of ~ b). Another remarkable property of the nonchiral bent-core molecules is the appearance of phases which probably possess some kind of helical superstructure [91]. Detailed clari®cation of the structures of these phases, as well as the characterization of their properties, are promising tasks for future research.

References [1] R.B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, J. Phys. (Paris) Lett. 36, L69 (1975). [2] R.B. Meyer, Mol. Cryst. Liq. Cryst. 40, 33 (1977). [3] Review articles and books on various aspects of ferroelectric liquid crystals and related topics are: S.T. Lagerwall, B. Otterholm, and K. Skarp, Material properties of ferroelectric liquid crystals and their relevance for applications and devices, Mol. Cryst. Liq. Cryst. 152, 503 (1987); S.T. Lagerwall, N.A. Clark, J. Dijon, and J.F. Clerc, Ferroelectric liquid crystals: The development of devices, Ferroelectrics 94, 3 (1989); J.W. Goodby, R. Blinc, N.A. Clark, S.T. Lagerwall, M. Osipov, S.A. Pikin, T. Sakurai, K. Yoshino, and B. ZÏeksÏ, Ferroelectric Liquid CrystalsÐPrinciples, Properties, and Applications, Gordon and Breach, New York, 1991; D.M. Walba, Ferroelectric liquid crystalsÐA unique state of matter, in: Advances in the Synthesis and Reactivity of Solids, Vol. 1, p. 173, JAI Press, London, 1991; T.P. Rieker and N.A. Clark, The layer and director structures of ferroelectric liquid crystals, NATO ASI Ser., Ser. B, 290 (Phase Transitions Liq. Cryst.), 287 (1992);

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

247

R. Blinc, Models for Phase transitions in ferroelectric liquid crystals: Theory and experimental results, NATO ASI Ser., Ser. B, 290 (Phase Transitions Liq. Cryst.), 343 (1992); J.W. Goodby, A.J. Slaney, C.J. Booth, I. Nishiyama, J.D. Vuijk, P. Styring, and K.J. Toyne, Chirality and frustration in ordered ¯uids, Mol. Cryst. Liq. Cryst. 243, 231 (1994); S.J. Elston, The optics of ferroelectric liquid crystals, J. Mod. Opt. 42, 19 (1995); I. MusÏevicÏ, B. ZÏeksÏ, R. Blinc, and Th. Rasing, Ferroelectric liquid crystals: From the plane wave to the multisoliton limit, Int. J. Mod. Phys. B 9, 2321 (1995); L.M. Blinov, Electric ®eld e¨ects in liquid crystals, in: Handbook of Liquid Crystal Research (edited by P.J. Collings and J.S. Patel), p. 125, Oxford University Press, New York, 1997; S.T. Lagerwall, Ferroelectric liquid crystals, in: Handbook of Liquid Crystals (edited by D. Demus, J.W. Goodby, G.W. Gray, H.-W. Spiess, and V. Vill), Vol. 2B, p. 515, Wiley-VCH, Weinheim, 1998. P.S. Pershan, The Structure of Liquid Crystal Phases, World Scienti®c, Singapore, 1988. G.W. Gray and J.W. Goodby, Smectic Liquid Crystals, Leonard Hill, Glasgow, 1984. J. Als-Nielsen, J.D. Litster, R.J. Birgeneau, M. Kaplan, C.R. Sa®nya, A. Lindegaard-Andersen, and S. Mathiesen, Phys. Rev. B 22, 312 (1980). Ch. Bahr and G. Heppke, Mol. Cryst. Liq. Cryst. 150b, 313 (1987). B.R. Ratna, R. Shashidhar, G.G. Nair, S.K. Prasad, Ch. Bahr, and G. Heppke, Phys. Rev. A 37, 1824 (1988). P.G. de Gennes, Mol. Cryst. Liq. Cryst. 21, 49 (1973). C.C. Huang and J.M. Viner, Phys. Rev. A 25, 3385 (1982). R.J. Birgeneau, C.W. Garland, A.R. Kortan, J.D. Litster, M. Meichle, B.M. Ocko, Ch. Rosenblatt, L.J. Yu, and J.W. Goodby, Phys. Rev. A 27, 1251 (1983). Y. Galerne, J. Phys. (Paris) 46, 733 (1985). S.L. Arora, J.L. Fergason, and A. Saupe, Mol. Cryst. Liq. Cryst. 10, 243 (1970). W. Helfrich and C.S. Oh, Mol. Cryst. Liq. Cryst. 14, 289 (1971). W.Z. Urbach and J. Billard, C. R. Acad. Sci. (Paris), Ser. B, 274, 1287 (1972). B.I. Ostrovski, A.Z. Rabinovich, A.S. Sonin, B.A. Strukov, and S.A. Taraskin, Ferroelectrics 20, 189 (1978). I. MusÏevicÏ, B. ZÏeksÏ, R. Blinc, L. Jansen, A. Seppen, and P. Wyder, Ferroelectrics 58, 71 (1984). R.B. Meyer, Phys. Rev. Lett. 22, 918 (1969). C.Y. Young, R. Pindak, N.A. Clark, and R.B. Meyer, Phys. Rev. Lett. 40, 773 (1978). C. Rosenblatt, R. Pindak, N.A. Clark, and R.B. Meyer, Phys. Rev. Lett. 42, 1220 (1979). B. Urbanc and B. ZÏeksÏ, Liq. Cryst. 5, 1075 (1989). P. Pieranski, E. Guyon, and P. Keller, J. Phys. (Paris) 36, 1005 (1975). F. Kremer, S.U. Vallerien, H. Kapitza, R. Zentel, and E.W. Fischer, Phys. Rev. A 42, 3667 (1990). J.R. Lalanne, J. Buchert, C. Destrade, H.T. Nguyen, and J.P. Marcerou, Phys. Rev. Lett. 62, 3046 (1989).

248

C. Bahr

[25] J.R. Lalanne, C. Destrade, H.T. Nguyen, and J.P. Marcerou, Phys. Rev. A 44, 6632 (1991). [26] T. Sakurai, N. Mikami, R. Higuchi, M. Honma, M. Ozaki, and K. Yoshino, J. Chem. Soc., Chem. Commun. 978 (1986). [27] Ch. Bahr and G. Heppke, Mol. Cryst. Liq. Cryst. Lett. 4, 31 (1986). [28] K. Mohr, S. KoÈhler, K. Worm, G. Pelzl, S. Diele, H. Zaschke, D. Demus, G. Andersson, I. Dahl, S.T. Lagerwall, K. Skarp, and B. Stebler, Mol. Cryst. Liq. Cryst. 146, 151 (1987). [29] N.A. Clark and S.T. Lagerwall, Ferroelectrics 59, 25 (1984). [30] J.W. Goodby, E. Chin, T.M. Leslie, J.M. Geary, and J.S. Patel, J. Am. Chem. Soc. 108, 4729 (1986). [31] J.W. Goodby and E. Chin, J. Am. Chem. Soc. 108, 4736 (1986). [32] D.M. Walba, S.C. Slater, W.N. Thurmes, N.A. Clark, M.A. Handschy, and F. Supon, J. Am. Chem. Soc. 108, 5210 (1986). [33] R. Bartolino, J. Doucet, and G. Durand, Ann. Phys. 3, 389 (1978). [34] E.N. Keller, E. Nachaliel, D. Davidov, and C. BoȨel, Phys. Rev. A 34, 4363 (1986). [35] N. Mikami, R. Higuchi, T. Sakurai, M. Ozaki, and K. Yoshino, Jpn. J. Appl. Phys. 25, L833 (1986). [36] J.W. Goodby, E. Chin, J.M. Geary, J.S. Patel, and P.L. Finn, J. Chem. Soc., Faraday Trans. 1, 83, 3429 (1987). [37] S. Dumrongrattana, C.C. Huang, G. Nounesis, S.C. Lien, and J.M. Viner, Phys. Rev. A 34, 5010 (1986). [38] B. ZÏeksÏ, Mol. Cryst. Liq. Cryst. 114, 259 (1984). [39] T. Carlsson, B. ZÏeksÏ, A. Levstik, C. FilipicÏ, I. Levstik, and R. Blinc, Phys. Rev. A 36, 1484 (1987). [40] T. Carlsson, B. ZÏeksÏ, C. FilipicÏ, A. Levstik, and R. Blinc, Mol. Cryst. Liq. Cryst. 163, 11 (1988). [41] T. Carlsson, B. ZÏeksÏ, C. FilipicÏ, and A. Levstik, Mol. Cryst. Liq. Cryst. 151, 155 (1987). [42] R. Meister and H. Stegemeyer, Ber. Bunsenges. Phys. Chem. 97, 1242 (1993). [43] R. Blinc and B. ZÏeksÏ, Phys. Rev. A 18, 740 (1978). [44] C. FilipicÏ, T. Carlsson, A. Levstik, B. ZÏeksÏ, R. Blinc, F. Gouda, S.T. Lagerwall, and K. Skarp, Phys. Rev. A 38, 5833 (1988). [45] S.M. Khened, S.K. Prasad, B. Shivkumar, and B.K. Sadashiva, J. Phys. France II 1, 171 (1991). [46] T. Carlsson, B. ZÏeksÏ, C. FilipicÏ, and A. Levstik, Phys. Rev. A 42, 877 (1990). [47] N.A. Clark and S.T. Lagerwall, Appl. Phys. Lett. 36, 899 (1980). [48] T.P. Rieker, N.A. Clark, G.S. Smith, D.S. Parmar, E.B. Sirota, and C.R. Sa®nya, Phys. Rev. Lett. 59, 2658 (1987). [49] N.A. Clark and T.P. Rieker, Phys. Rev. A 37, 1053 (1988). [50] See, e.g., W.J.A.M. Hartmann, Ferroelectrics 122, 1 (1991). [51] G. Moddel, K.M. Johnson, W. Li, R.A. Rice, L.A. Pagano-Stau¨er, and M.A. Handschy, Appl. Phys. Lett. 55, 537 (1989). [52] M.R. Meadows, M.A. Handschy, and N.A. Clark, Appl. Phys. Lett. 54, 1394 (1989). [53] M. Ozaki, Y. Sadohara, T. Hatai, and K. Yoshino, Jpn. J. Appl. Phys. 29, 843 (1990). [54] J.S. Patel, Appl. Phys. Lett. 60, 280 (1992).

8. Smectic Liquid Crystals: Ferroelectric Properties and Electroclinic E¨ect

249

[55] L.A. Beresnev, V.G. Chigrinov, D.I. Dergachev, E.I. Poshidaev, J. FuÈnfschilling, and M. Schadt, Liq. Cryst. 5, 1171 (1989). [56] L.A. Beresnev, W. Haase, A.P. Onokhov, W. Dultz, M.V. Isaev, N.A. Feoktistov, N.L. Ivanova, E.A. Konshina, A.N. Chaika, V.A. Berenberg, and T. Weyrauch, Mater. Res. Soc. Symp. Proc. 488, 859 (1998). [57] L.A. Beresnev, J. Hossfeld, W. Dultz, A.P. Onokhov, and W. Haase, Mater. Res. Soc. Symp. Proc. 413, 363 (1996). [58] S. Garo¨ and R.B. Meyer, Phys. Rev. Lett. 38, 848 (1977). [59] S. Garo¨ and R.B. Meyer, Phys. Rev. A 19, 338 (1979). [60] G. Andersson, I. Dahl, P. Keller, W. Kuczynski, S.T. Lagerwall, K. Skarp, and B. Stebler, Appl. Phys. Lett. 51, 640 (1987). [61] Ch. Bahr and G. Heppke, Liq. Cryst. 2, 825 (1987). [62] G. Andersson, I. Dahl, L. Komitov, S.T. Lagerwall, K. Skarp, and B. Stebler, J. Appl. Phys. 66, 4983 (1989). [63] S. Nishiyama, Y. Ouchi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 26, L1787 (1987). [64] S.-D. Lee and J.S. Patel, Appl. Phys. Lett. 54, 1653 (1989). [65] Ch. Bahr and G. Heppke, Phys. Rev. A 41, 4335 (1990). [66] F. Gieûelmann and P. Zugenmaier, Phys. Rev. E 52, 1762 (1995). [67] L. Ruan, J.R. Sambles, and J. Seaver, Liq. Cryst. 19, 133 (1995). [68] A. Michelson and L. Benguigui, Phys. Rev. A 18, 2736 (1978). [69] S. Garo¨ and R.B. Meyer, Phys. Rev. A 18, 2739 (1978). [70] Ch. Bahr and G. Heppke, Phys. Rev. Lett. 65, 3297 (1990). [71] L. Dupont, M. GlogarovaÂ, J.P. Marcerou, H.T. Nguyen, C. Destrade, and L. LejcÏek, J. Phys. II France 1, 831 (1991). [72] F. Gieûelmann, A. Heimann, and P. Zugenmaier, Ferroelectrics 200, 237 (1997). [73] Ch. Bahr, G. Heppke, and B. Sabaschus, Liq. Cryst. 11, 41 (1992). [74] Ch. Bahr, G. Heppke, and B. Sabaschus, Ferroelectrics 84, 103 (1988). [75] H.-R. DuÈbal, C. Escher, and D. Ohlendorf, Ferroelectrics 84, 143 (1988). [76] Ch. Bahr, G. Heppke, and B. Sabaschus, Liq. Cryst. 9, 31 (1991). [77] M. GlogarovaÂ, Ch. Destrade, J.P. Marcerou, J.J. Bonvent, and H.T. Nguyen, Ferroelectrics 121, 285 (1991). [78] V.V. Ginzburg, R. Shao, N.A. Clark, and D.M. Walba, Proc. SPIE-Int. Soc. Opt. Eng. 2175, 102 (1994). [79] H.Y. Liu, C.C. Huang, Ch. Bahr, and G. Heppke, Phys. Rev. Lett. 61, 345 (1988). [80] H.Y. Liu, C.C. Huang, T. Min, M.D. Wand, D.M. Walba, N.A. Clark, Ch. Bahr, and G. Heppke, Phys. Rev. A 40, 6759 (1989). [81] S.K. Prasad, V.N. Raja, D.S. Shankar Rao, G.G. Nair, and M.E. Neubert, Phys. Rev. A 42, 2479 (1990). [82] J.J. Bonvent, Ch. Destrade, M. GlogarovaÂ, J.P. Marcerou, and H.T. Nguyen, Ferroelectrics 145, 213 (1993). [83] W. Kuczynski and H. Stegemeyer, Chem. Phys. Lett. 70, 123 (1980). [84] H. Stegemeyer, R. Meister, U. Ho¨mann, A. Sprick, and A. Becker, J. Mater. Chem. 5, 2183 (1995). [85] H. Stegemeyer, A. Sprick, M.A. Osipov, V. Vill, and H.-W. Tunger, Phys. Rev. E 51, 5721 (1995). [86] H.R. Brand, P.E. Cladis, and H. Pleiner, Macromolecules 25, 7223 (1992). [87] T. Akutagawa, Y. Matsunaga, and K. Yasuhara, Liq. Cryst. 17, 659 (1994).

250

C. Bahr

[88] E.A.S. Bustamante, S.V. Yablonskii, B.I. Ostrovskii, L.A. Beresnev, L.M. Blinov, and W. Haase, Liq. Cryst. 21, 829 (1996). [89] T. Niori, T. Sekine, J. Watanabe, T. Furukawa, and H. Takezoe, J. Mater. Chem. 6, 1231 (1996). [90] D.R. Link, G. Natale, R. Shao, J.E. Maclennan, N.A. Clark, E. KoÈrblova, and D.M. Walba, Science 278, 1924 (1997). [91] G. Heppke and D. Moro, Science 279, 1872 (1998).

9

Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases Hideo Takezoe and Yoichi Takanishi

9.1

Introduction

New materials are discovered sometimes intentionally and sometimes accidentally. The former case can be exempli®ed by ferroelectric liquid crystals (FLCs), wherein the chirality was introduced into the lateral chain to reduce the symmetry, leading to a noncentrosymmetric system [1]. In the latter case, it can be referred to as antiferroelectric liquid crystals (AFLCs) which were discovered accidentally. Actually, compounds which exhibit the antiferroelectric phase had been synthesized several years before they were proven as AFLCs. The hitherto known AFLCs include three materials, as shown in Figure 9.1: (1) MHPOBC, (2) MHTAC, and (3) (R) and (S)-1-methylpentyl 4 0 -(4 00 -n-decyloxybenzoyloxy)bipheny-l-4-carboxylates. MHPOBC was ®rst reported as a new FLC in the Japan domestic liquid crystal meeting in 1985 by Inukai et al. [2]. In the First International FLC Conference in 1987, two groups pointed out unusual behavior in this compound. Hiji et al. [3] reported a third stable state exhibiting a dark view between crossed polarizers when one of the polarizers is parallel to the smectic layer. Furukawa et al. [4] reported a very small dielectric constant and a threshold behavior in the electro-optic response in the lower temperature region of SmC*, suggesting a new phase, SmY*. The MHTAC analogue was ®rst synthesized in 1976, and MHTAC itself appeared in the literature in 1983 [5]. However, the herring-bone (antiferroelectric) structure of this material was ®rst reported in the Second International FLC Conference (1989, Goteborg) [6]. At the same conference, Takezoe et al. also reported the antiferroelectricity of MHPOBC [7], [8]. Goodby and Chin [9] also synthesized and reported (R) and (S)-1-methylpentyl 4 0 -(4 00 -n-decyloxybenzoyloxy)biphenyl-4-carboxylates in 1988 without noticing the existence of the antiferroelectric phase in these compounds. Thus, an induction period was necessary to identify the antiferroelectric structure after the synthesis of the AFLC materials and the discovery of some characteristic features. This period was required to justify the incompatibility of the antiferroelectric structure with the molecular arrangement in 251

252

H. Takezoe and Y. Takanishi

Figure 9.1. Chemical structures of three antiferroelectric compounds reported without realizing them as antiferroelectric ones.

liquid crystals, in which the herring-bone molecular arrangement is strongly ruled out in liquid crystals from the viewpoint of packing entropy or excluded volume e¨ect. Actually, all the molecular arrangements in the other liquid crystalline systems composed of rod-like molecules are such that the neighboring molecules tend to be parallel to each other. The most important observation of AFLCs was made in 1988 before the identi®cation of the antiferroelectric phase; i.e., tristable switching with a sharp threshold and a double hysteresis [10]. The tristable switching is observable by two methods, i.e., the electro-optic e¨ect and the switching current measurements. These results are shown in Figure 9.2 [10], in which

Figure 9.2. Transmittance change and switching current observed by applying a triangular voltage wave in MHPOBC [10]. Note that two changes in the transmittance and two peaks in the switching current are observed.

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

253

Figure 9.3. Apparent tilt angle under the application of a dc electric ®eld in MHPOBC [10]. Note the threshold and double hysteresis behaviors.

two switching current peaks appear when sharp transmittance changes occur. Figure 9.3 shows the apparent tilt angle under the application of a dc electric ®eld [10]. It is easily noticed that a switching device could be made by utilizing the threshold and the hysteresis. Figure 9.4 demonstrates the transmittance changes by applying positive and negative pulses superposed on a biased voltage [10]. This is the essential principle of the tristable AFLC display [11].

Figure 9.4. Electro-optic response by applying positive and negative pulses superposed on a biased voltage (6 V) in MHPOBC [10].

254

H. Takezoe and Y. Takanishi

9.2

Experimental Observations Indicating the Antiferroelectric Structure

The helicoidal structures of FLCs and AFLCs are shown in Figures 9.5(a) and (b), respectively. In the antiferroelectric structure, the helicoidal pitch is normally of the order of 1 mm, so that several hundreds of layers exist in a pitch length. Therefore, the molecules in the neighboring layers tilt almost to the opposite directional sense. The local molecular arrangement in the AFLCs is shown in Figure 9.5(c). At least four experimental observations, which strongly suggest or certify the antiferroelectric structure, were made. They are: (1) (2) (3) (4)

transmittance spectra in oblique incidence; interferential microscope observation of a droplet; ellipsometry for free-standing ®lms; and microscope observation of defects.

It is easy to understand the di¨erence of the optical properties of the ferroelectric and antiferroelectric helicoidal structures shown in Figures 9.5(a) and (b). For light propagating along the helicoidal axis, the two structures are apparently the same, since half the pitch is optically one period in both structures. In contrast, for obliquely incident light, the molecular orientations

Figure 9.5. Helicoidal structures of the ferroelectric and antiferroelectric phases, and a local molecular arrangement in SmCA .

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

255

in the layers separated by half the pitch look di¨erent, so that the optical period is a full pitch in the ferroelectric structure, while the optical period is still half the structural pitch in the antiferroelectric structure. Because of the herring-bone structure in the antiferroelectric phase, the local system is strongly biaxial and the major axis of the indecatrix is parallel to the layer normal and the two optic axes rotate about the layer normal from layer to layer due to the macroscopic helicoidal structure. Contrary to this structure, the indecatrix of the ferroelectric structure is weakly biaxial due to hindered rotation, and the major axis of the indecatrix tilts and precesses from layer to layer. These two di¨erent structures can be easily di¨erentiated by observing the selective re¯ection at oblique incidence [12]. Namely, a so-called full pitch band is observable in the ferroelectric phase at the wavelength of twice of that of the normal selective re¯ection band, while the full pitch band must be absent in the antiferroelectric phase. This is actually the case based on the ®rst experimental evidence, as shown in Figure 9.6 [12], in which no full

Figure 9.6. Transmittance spectra at oblique incidence in the ferroelectric and the antiferroelectric phases of MHPOBC [12].

256

H. Takezoe and Y. Takanishi

pitch band can be seen at arrow positions. This presence of antiferroelectric phase can be inferred from Figure 9.6. The herring-bone molecular arrangement in the SmO phase of MHTAC was ®rst con®rmed by an interferential microscope observation of a droplet of racemic MHTAC on a glass surface [13]. The temperature of the droplet was controlled so that a few smectic layers were formed on the isotropic droplet. By changing the temperature under the application of an electric ®eld parallel to the glass surface, the number of layers was changed and the structure was examined. In this way, it was con®rmed that the molecular tilt sense alters from layer to layer. It has been con®rmed that the SmO and SmCA phases are identical [14]±[16]. The local molecular arrangement, i.e., herring-bone structure, was also con®rmed by an ellipsometry using free-standing ®lms. The optical geometry used is shown as an inset of Figure 9.7. The phase di¨erence D ˆ D p ÿ Ds between the p and s components of the obliquely incident transmitted light was measured in free-standing ®lms, and is shown in Figure 9.7. The phase di¨erences under positive and negative electric ®elds, D ‡ and D ÿ , are always

Figure 9.7. Phase di¨erence between s and p components of the transmitted light measured under positive and negative ®elds in free-standing ®lms [17].

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

257

Figure 9.8. Schlieren texture observed in a homeotropic cell of the SmCA phase of MHPOBC. Note the existence of two-brush defects.

di¨erent in the SmC* phase, since the molecules tilt either toward or away from the incident laser beam. In the SmCA phase, however, D ‡ ˆ D ÿ for two-layer ®lms, while D ‡ is not equal to D ÿ for three-layer ®lms. This experiment clearly reveals the layer-by-layer alternation of the tilt and polarization direction. To identify the antiferroelectric phase, texture observation of the homeotropic cells of racemic compounds is very e¨ective. In the SmC phase, only the schlieren texture with four brushes is observable and that with two brushes is prohibited, because of the head-and-tail inequivalence of the Cdirector. In the SmCA phase, however, the schlieren texture with two brushes is sometimes seen, as shown in Figure 9.8 [18], [19]. The existence can be explained by taking into account a screw dislocation, as illustrated in Figure 9.9. The discontinuous change (p-wall) of the C-director is compensated by the screw dislocation. This defect is a combined defect of a disclination and a dislocation, i.e., a dispiration [18], [19]. The experiments mentioned above could be used to identify the SmCA phase. Other methods to identify the SmCA phase are the observation of the tristable switching [10], conoscope observations [20], [21], and dielectric measurements [22], [23]. However, the tristable switching observable by the electro-optic and the switching current measurements are not enough to identify the SmCA phase. Actually it was shown that FLCs with a very short pitch are possible to exhibit the tristable switching, since helicoidal structure is extremely stabilized against an applied electric ®eld (helicoid stabilized FLCs) [24].

258

H. Takezoe and Y. Takanishi

Figure 9.9. Model structure of the two-brush defect; i.e., dispiration, a combined defect of a wedge disclination and a screw dislocation.

9.3

Microscopic Origin of Antiferroelectricity

Generally, molecules tend to align parallel to each other in liquid crystalline phases. Therefore, we ought to seek some particular causes for the appearance of the SmCA phase. The preliminary experimental observations made in this phase are summarized as follows: (1) The SmCA phase exists in compounds, the optical isomer of which have large spontaneous polarization [25]. (2) The SmCA phase exists even in racemic compounds which have no spontaneous polarization [26]±[30]. (3) There exist compounds which show SmC* and SmCA depending on chiral end chain length; odd±even behavior in the appearance of the SmCA phase [28], [31]. The origin of the formation of the antiferroelectric structure has not yet been clari®ed. The existing ideas comprise of a dipole pairing model [32], a

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

259

Figure 9.10. Chemical structure of a swallow-tailed liquid crystal compound.

Px model [33], and a steric interaction model [34]. The importance of a steric interaction between the end chains of molecules in adjacent layers is suggested particularly, because even nonchiral molecules with swallow-tailed end chains of the same length (Figure 9.10) exhibit the antiferroelectric, more strictly anticlinic, molecular orientation [34]. The experimental observation (3) also suggests the importance of the steric interaction. Figure 9.11 shows the phase behavior in three homologous series of smetogens [31]. TFMHPOBC (see Figure 9.18) homologues have a strong tendency of antiferroelectricity and exhibit only the SmCA phase. However, MHPOBC homologues and ¯uorine-substituted TFMHPOBC homologues show the odd±even e¨ect. The experimental observation (1) indicates the importance of an electric interaction. The phase behavior in the dimer liquid crystal shown in Figure 9.12 [35], [36] is very suggestive. This type of liquid crystal with an alkylene spacer is known to exhibit the conformation shown in Figure 9.13 [37]; two mesogenic groups are parallel to each other when the number of carbons of the spacer group, m, is even, while they take an anticlinic orientation for odd m. In the liquid±crystal homologues shown in Figure 9.12, this is actually the case for m less than 9, i.e., the phase sequences are Iso±SmA±SmC* for even m and Iso±SmCA for odd m. However, for m ˆ 10 and 12, the phase sequence Iso±SmA±SmCA is realized even for even m [35]. This result can be attributed to an electric interaction stabilizing the antiferroelectric structure, which overcomes the regularity due to the conformation, by increasing the ¯exibility of the spacer group. The dipole pairing model is based on the pair formation of molecules in adjacent layers through the dipole±dipole interaction. Because of the experimental observation (2), pairing must be made between like enantiomers, as shown in Figure 9.14 [32]. Otherwise, anticlinic orientation cannot be formed in the racemic compounds. Therefore, in this model, chiral molecular recognition is required. The pairing may be dynamic and may occur in optically resolved local enantiomeric domains. The so-called Px model is related to the polarizations existing at layer boundaries. The SmCA phase has D2 symmetry with two C2 -axes; i.e., one at the middle of the layer and perpendicular to the molecular tilt plane (Py ) and one at the layer boundary parallel to the tilt plane (Px ), as shown in Figure 9.15. Then, Px ±Px interaction between two adjacent layer boundaries stabilizes the antiferroelectric structure, if the ¯uctuation of Px is involved. Even in racemic compounds, Px always exists, though Py is zero. Thus, this model

260

H. Takezoe and Y. Takanishi

Figure 9.11. Phase behavior of the three homologous series of antiferroelectric liquid crystals.

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

261

Figure 9.12. Chemical structures and phase sequences of dimer liquid crystal com-

satis®es the experimental observation (2). Actually, polarized FT±IR absorption of the carbonyl group (CbO) gives di¨erent polarization characteristics in SmC* and SmCA , leading to di¨erent orientation of CbO, i.e., it rather lies on the tilt plane in SmCA while it takes a considerably upright position in SmC* [33]. The analysis was made without taking account the di¨erent degrees of hindrance of the molecular motion about its long axis in the two phases. However, Px always does exist irrespective of the CbO direction from the symmetry viewpoint and has an in¯uence on the stabilization of the antiferroelectric phase. At this moment, it is still an open question as to which of the pairings or Px interactions plays a dominant role in stabilizing the antiferroelectric phase. In any event, for these electrostatic models to be e¨ective, it is required that the dipoles in adjacent layers must be su½ciently close to each other. The

262

H. Takezoe and Y. Takanishi

Figure 9.13. Two possible orientations of two mesogenic groups of the dimer liquid crystals.

Figure 9.14. Possible situation of dipole pairing in a partially racemized antiferroelectric liquid crystal. Only like enantiomers make pairs.

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

263

Figure 9.15. Two kinds of C2 -axes, along which polarization can arise; i.e., parallel and perpendicular to the tilt plane.

dipole moment responsible for the spontaneous polarization locates near the chiral center. Therefore, if the shape of the molecules is more or less straight or of a zig-zag shape, the distance between dipoles in adjacent layers will be Ê , leading to less e¨ective dipole±dipole interaction. more than 10 A The situation would be di¨erent if molecules were bent at the chiral position. The bent molecular con®guration was actually con®rmed in the crystalline phase of MHPOBC by X-ray di¨raction [38], [39]; the chiral end chain is bent by about 90 with respect to the core direction. It was also con®rmed, at least by three experiments, that the molecules are also bent even in the liquid crystalline phase. According to NMR [40], [41] and polarized FT±IR [42] measurements, the chiral chain is bent at least by the magic angle (54:7 ) with respect to the core. The layer spacing of MHPOBC analogues, in which the CH3 group at the chiral group is substituted by C r H2r‡1 (r ˆ 2 @ 6), increases with increasing r, suggesting the bent molecular con®guration shown in Figure 9.16 [43]. Finally, the postulated molecular con®guration is shown in Figure 9.17. Nevertheless, it is still necessary to con®rm whether most of the antiferroelectric molecules are bent or not and whether the molecules showing only the ferroelectric phase are bent or not.

9.4 9.4.1

Ferrielectric and Antiferroelectric Subphases Experimental Observation of Subphases

One of the most striking features of AFLCs is the emergence of various subphases. The discovery of the subphases was associated with the discovery

Figure 9.16. Temperature-dependence of the layer spacing of a homologous series of MHPOBC derivatives in which the CH3 group at the chiral position is substituted by C r H2r‡1 …r ˆ 2 @ 6† [43]. The molecular structures obtained by MOPAC are also shown.

264 H. Takezoe and Y. Takanishi

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

265

Figure 9.17. Molecular con®guration of MHPOBC in SmA suggested by NMR, FT±IR, and X-ray analysis.

of the antiferroelectric phase in MHPOBC, which exhibits the SmCa  phase between SmA and SmC* and the SmCg  phase between SmC* and SmCA [12]. Later a reentrant antiferroelectric phase, called the AF phase, was discovered in MHPBC (see Figure 9.18) between SmCa  and SmCg  [44]. Ferroelectric and antiferroelectric compounds possessing a similar core structure are summarized in Figure 9.18. These compounds are arranged almost in order of the strength of their ferroelectricity. Namely upper and

266

H. Takezoe and Y. Takanishi

Figure 9.18. Ferroelectric and antiferroelectric compounds with a similar core structure.

lower compounds have stronger ferroelectricity and antiferroelectricity, respectively, and exhibit only the ferroelectric and antiferroelectric phases, respectively. In MHPOOCBC, SmCA is hidden by crystallization and only the subphase SmCg  appears. Moreover, the temperature range of SmC* is fairly wide. Thus, MHPOOCBC has relatively high ferroelectricity and weak

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

267

antiferroelectricity. In MHPBC, on the other hand, the SmCa  and AF phases appear in addition to SmCg  , and SmC* disappears, indicating that this material has relatively high antiferroelectricity. When the antiferroelectricity becomes more intense, subphases except for SmCa  disappear (MHPOCBC and TFMHPBC) and ®nally all the subphases disappear and the direct transition from SmA to SmCA occurs (EHPOCBC, TFMHPOBC, and TFMHPOCBC). The phase transitions can be identi®ed by many techniques such as a thermal measurement [45]±[48], microscope observation, a dielectric measurement [22], and a layer spacing measurement by X-ray di¨raction [31]. A much advanced technique such as ellipsometry [17] and the observation of freely suspended ®lms under temperature gradient [49] could also be utilized. However, the identi®cation of these subphases is not easy. Isozaki et al. [50]± [53] have adopted mainly conoscopic observations using homeotropic cells or freely suspended ®lms [20], [21]. The conoscopic ®gures in all the phases exhibit a uniaxial pattern in the absence of an electric ®eld, although the center of the extinction cross is not completely dark because of the rotatory power due to helicoidal structures. Therefore, the conoscopic observations to identify phases must be made under an electric ®eld. Figure 9.19 shows typical conoscopic ®gures in the SmCa  , SmC*, SmCg  , and SmCA phases under an appropriate electric ®eld. All of them exhibit biaxial ®gures characteristic of the respective phases. The richness of subphases is recognized by mixing two compounds. The richness is enhanced when two compounds have moderate strength of antiferroelectricity and ferroelectricity. In order to make phase diagrams of binary mixtures, phase diagrams of temperature versus electric ®eld (T±E phase diagram) have to be made for several binary mixtures. An example is shown in Figure 9.20, in which the mixtures of two compounds (EHPOCBC and MHPOOCBC) were used [51]. Using these T±E phase diagrams, the phase diagram of temperature versus wt% (x) of EHPOBC (T±x phase diagram) is drawn, as shown in Figure 9.21(a) [51]. Other T±x phase diagrams in Figure 9.21 are for some other binary mixtures [50], [51], [53]. In this way, at least ®ve subphases have been suggested to appear between SmC* and SmCA ; i.e., SmC*±FI±AF±FIH ±SmCg  ±FIL ±SmCA even though they are not always observable. However, if they exist, they do appear with the same order. The T±x phase diagrams of isomers with various optical purities are also shown in Figure 9.22 [52]. At ®rst, it was thought that the SmCa  phase exists only in enantiomers with high optical purity [26], [46], as shown in Figures 9.22(a) and (b); and the SmC* phase is injected or becomes wider with decreasing optical purity, as shown in Figures 9.22(a) and (c). However, SmCa  is injected by lowering the optical purity in TFMHPOCBC (Figure 9.22(d)) and SmC* is not injected even in racemate (Figure 9.22(b)). All the phase behavior could be interpreted by taking into account the competing interaction of ferroelectricity and antiferroelectricity [54].

268

H. Takezoe and Y. Takanishi

Figure 9.19. Typical conoscopic ®gures, indices of ellipsoid, and molecular arrangements in the SmCA , SmCg  , and SmC* phases under appropriate electric ®elds.

9.4.2

Theoretical Treatment

The theoretical treatment for explaining the structures of the subphases is classi®ed into two, i.e., the Ising model and XY model. In the Ising model, molecules locally lay on a plane and tilt right or left, while they distribute at any azimuthal angles on a cone in the XY model. From the above discussion, it is evident that there exist competing interactions which stabilize ferroelectricity and antiferroelectricity; i.e., the excluded volume e¨ect stabilizes a ferroelectric (synclinic) structure and the dipole±dipole interaction or steric interaction stabilizes an antiferroelectric (anticlinic) structure. We discussed the appearance of various subphases based on the one-dimensional Ising model with long-range repulsive interactions developed by Bak and Bruinsma [55], [56]. Namely, because of the competing interaction of

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

269

Figure 9.20. T±E phase diagrams of binary mixtures of EHPOCBC and MHPOOCBC with various mixing ratios: (a) (R)-EHPOCBC; (b) 50.0% (R)-EHPOCBC in (R)-MHPOOCBC; (c) 19.8% (R)-EHPOCBC in (R)-MHPOOCBC; and (d) 18.8% (R)-EHPOCBC in (R)-MHPOOCBC.

ferroelectricity and antiferroelectricity, ferroelectric (F) ordering is excited in antiferroelectric (A) ordering and the ferroelectrically ordered positions distribute uniformly due to a long-range repulsion interaction between the F orderings [54]. Since the fraction of F ordering is possible for every rational number q, an in®nite number of subphases may appear, the devil's staircase [57]. According to this model, the major subphases, SmCg  and AF, are described by the structures shown in Figure 9.23. Yamashita and Miyazima [58] and Yamashita [59], [60] adopted the modi®ed ANNNI (axial next-nearest-neighbor Ising) model with the thirdneighboring interaction (ANNNI ‡ J3 model) and wrote the Hamiltonian X X X X si sj ÿ J1 si si‡1 ÿ J2 si si‡2 ÿ J3 si si‡3 ; …9:1† H ˆ ÿJ …i; j†

i

i

i

where the Ising spin si takes a value of G1 corresponding to the molecular tilting senses of the ith smectic layer. The ®rst summation is taken over nearest-neighboring pairs …i; j† in the same smectic layer, and other summa-

270

H. Takezoe and Y. Takanishi

Figure 9.21. T±x phase diagrams of: (a) (R)-EHPOCBC and (R)-MHPOOCBC; (b) (R)-MHPOCBC and (R)-MHPOOCBC; (c) (R)-MHPBC and (R)-TFMHPBC; and (d) (R)-MHPBC and (S)-TFMHPBC.

tions are only for the ®rst, second, and third neighboring pairs in the axial direction parallel to the layer normal; the second-nearest-neighbor interaction J2 should be negative to ensure competition, and the third-nearestneighbor interaction J3 , which is positive or negative, is included for the possible wide stability of SmCg  . Yamashita [61], [62] also pointed out an important role played by the directional sense of the molecular long axis and gave realistic physical grounds for these rather long-range interactions. A similar treatment has been adopted by Koda and Kimura [63]. Yamashita showed that four ground states appear, i.e., SmCA …q ˆ 12†, SmCg  …q ˆ 13†, AF …q ˆ 14†, and SmC* …q ˆ 0†, as illustrated in Figure 9.24.

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

271

Figure 9.22. T±x phase diagrams of isomers with various optical purity in: (a) MHPOBC; (b) MHPOCBC; (c) TFMHPOBC; and (d) TFMHPOCBC.

Figure 9.23. Molecular orientation structures of the SmCg  and AF phases based on the one-dimensional Ising model.

272

H. Takezoe and Y. Takanishi

Figure 9.24. Phase diagram obtained by the ANNNI model.

Some other subphases were also predicted at ®nite temperatures. Our treatment and ANNNI ‡ J3 model give essentially the same structures for the major subphases, SmCg  and AF, as displayed in Figure 9.23, although the corresponding rational numbers are di¨erent [49]. It is important to note that a macroscopic helix with about a 1 micron pitch is formed even in the Ising model because of the chirality introduced in molecules, so that this tilt plane rotates along the layer normal. The phenomenological Landau models have been developed by many authors by taking into account the coupling of ferroelectric and antiferroelectric order parameters in bilayer [64]±[68] and axially next-nearest-neighbor (ANNN) interaction [69], [70]. The bilayer models are impractical for SmCg  , since no layer spacing change along the layer normal has been observed. The XY character was introduced by considering ANNN interaction (ANNNXY model), leading to a molecular arrangement with very short pitch consisting of three or four layers (spiral model or clock model [69]± [71]; see Figure 9.25). Roy and Madhusudana [70] calculated the phase sequence using the ANNNXY model and succeeded in obtaining the phase sequence SmA±SmCa  ±SmC*±FIH -FII (SmCg  )±FIL ±SmCA . Moreover,

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

273

Figure 9.25. Molecular arrangements obtained by a spiral or clock model.

they calculated the molecular orientation under the application of an electric ®eld and showed that the clock model, consisting of three layers, is possible to have a metastable state and exhibit a conoscopic ®gure, which are the same as those observed in SmCg  [72]. Recently, Mach et al. [73] performed a resonant X-ray scattering using the free-standing ®lms of sulfur-containing AFLC materials possessing subphases and claimed that the clock model can explain the experimental results. The XY model gives helical structures with a very short pitch consisting of several layers. Because of such an ultrashort pitch, the system should appear to be optically uniaxial with the optic axis along the layer normal, and to exhibit negligible circular dichroism (CD) and optical rotatory power (ORP). However, macroscopic helices have been observed experimentally not only in SmCA [74] but also in the subphases. Actually, Miyachi et al. [75], [76] observed laser light di¨raction in the SmCg  and AF phases of (R)MHPBC. In this respect, the XY model, particularly the clock model, is not a realistic model for various subphases possessing macroscopic helices. Future studies are necessary to fully understand the appearance of various subphases. Many attempts to clarify the SmCa  phase have been made. Experimentally, the following characteristics are known: (1) It is a tilted phase [77], [78]. (2) Within the phase, various structural changes with temperature and an electric ®eld occur [32], [79], [80].

274

H. Takezoe and Y. Takanishi

(3) The helical structure may exist and the pitch length is either very short [78], [81] or very long [82]. (4) A very slow ¯uctuation mode was observed in dynamic light-scattering measurements [83]. (5) The SmA±SmCa  phase transition is of the second order and is close to the tricritical point [78], [84]. Future studies are necessary to clarify the structure.

9.5 9.5.1

Frustration Between Ferroelectricity and AntiferroelectricityÐV-Shaped Switching Discovery of V-Shaped Switching and Its Characteristics

As already mentioned, it is well established that many subphases such as antiferroelectric (AF) and ferrielectric (F) phases emerge under the competing interactions for stabilizing ferroelectricity and antiferroelectricity. This situation can be described as a competition between synclinic and anticlinic interactions, namely, the interlayer tilting correlation. A decrease in the degree of the tilting correlation or the frustration of ferroelectricity and antiferroelectricity brings about another interesting phenomenon; i.e., V-shaped switching characterized by thresholdless, hysteresis-free, and domainless behavior. For the V-shaped switching, Inui et al. [85] and Fukuda et al. [86] have suggested a phase with randomly oriented C-directors due to the reduction of the interlayer tilting correlation, without any piece of experimental evidence. The dynamic switching behaviors seemed to be explained by the random model based on the two-dimensional Langevin function [85]. However, the phase with randomly oriented C-directors has never yet been con®rmed. So far, only two mixtures composed of two and three compounds (Figure 9.26) showing V-shaped switching have been reported [85]±[94]. These mixtures show ambiguous phase sequences such as the ferrielectric phases over a wide temperature range in the bulk states [89], [90]. Since the ferrielectric phases are known to appear due to the competition (frustration) between interactions stabilizing ferroelectricity and antiferroelectricity, it is reasonable to consider that the phase, in which the interlayer correlation or the tilting correlation is reduced, manifests itself by V-shaped switching. Thin homogeneously aligned cells showing V-shaped switching exhibit a uniformly dark view in the absence of a ®eld under crossed polarizers, one of which is parallel to a smectic layer. If there are ferroelectric and/or ferrielectric domains, the size of which is larger than the wavelength of the visible light, one can see domains with di¨erent brightness. The evolution of switching characteristics from tristable to V-shaped was

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

275

Figure 9.26. Materials showing V-shaped switching, two mixtures/®ve compounds.

observed with increasing temperature and frequency in an apparently AFLC (compound 3 in Figure 9.26), as shown in Figure 9.27 [89]. In the electrooptic response at 23  C and at 0.1 Hz, the evolution is clearly seen because of the accidental di¨erence of the two substrate surface conditions: The stable state is the antiferroelectric phase. However, it is clear that the dark state at 0 V dynamically driven (arrow 1) has a di¨erent structure from the ordinary antiferroelectric structure (arrow 2) which also shows a dark view, since there is no reason that the same arrangement emerges twice in a quarterperiod of switching. Then, the questions are as follows: (1) What is the stable molecular orientation at 0 V; (2) What is the dynamics of the switching? and (3) What is the origin of the particular stable orientation at 0 V?

9.5.2

Collective Motion During V-Shaped Switching

Park et al. [92] measured the e¨ective optical anisotropy Dneff , the apparent tilt angle yapp , the switching current, and the second-harmonic generation in the three-component mixture (3:4:5 ˆ 40:40:20 (wt%) in Figure 9.26) and

276

H. Takezoe and Y. Takanishi

Figure 9.27. Transmittance versus electric ®eld on applying a triangular waveform of 0.1 Hz and 0.5 Hz at 23  C, 26  C, and 29  C in a homogeneously aligned cell of the compound 3 of Figure 9.26. Note that evolution is observed from tristable to V-shaped switching by changing the frequency and temperature.

compared them with the simulated results based on two extreme models, i.e., the random model and the collective model, where molecules switch randomly and collectively, respectively. The comparison revealed that the collective switching motion of liquid crystal molecules is more reasonable than the random motion, as will be shown in the following. Moreover, it was also con®rmed that the observed infrared absorption anisotropy of the phenyl stretching mode due to liquid crystal molecular distributions strongly supports the collective model. Figure 9.28 illustrates the electro-optic responses observed at two Y's; 0 and 30 , where Y is the angle between the layer normal and one of the crossed polarizer directions. As shown in Figure 9.28 [92], the electric ®eld dependence of transmittance at Y ˆ 0 shows the typical thresholdless, hysteresis free, V-shaped switching. From the electro-optic measurements at every 5 of Y, the transmittance T versus Y was obtained at given electric ®elds. Figure 9.29(a) shows three examples of T versus Y curves at applied ®elds of 0, ‡6, and ÿ6 V/mm [92]. The transmittance T is described by   pDneff d ; …9:2† T ˆ sin 2 …2…Y ‡ yapp †† sin 2 l where d is the cell thickness and l the light wavelength. Normally, the electro-optic transmittance change at ®xed temperatures is regarded as a function of only yapp …F †, where F is an applied electric ®eld. However, yapp is also a function of F, since yapp is determined by molecular distribution. Park et al. [92] analyzed Dneff as well as yapp as a function of F. As is easily seen in

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

277

Figure 9.28. The electrooptic responses observed in a homogeneous cell during the application of a triangular wave of 0.1 Hz for Y ˆ 0 and 30 .

(9.2), the amplitude and the phase of the T±Y curve give Dneff and yapp at a given electric ®eld, respectively. Figures 9.29(b) and (c) show yapp and Dneff of the V-shaped switching cell as a function of the applied electric ®eld, respectively. Simulation of yapp and Dneff was made for the two extreme models, i.e., the random and collective models. The results are shown by solid (collective) and broken (random) curves in Figure 9.29(c). It is clear that the calculated Dneff for the collective model with a layer tilt angle of 8 is the same as the experimental result, while the random model gives serious disagreement. From these results, it is clear that the collective behavior of liquid crystal molecules is more reasonable than the random distribution for the V-shaped switching. Very intuitive evidence of the molecular uniform orientation in the absence of a ®eld is given by polarized IR absorption [92]. Figure 9.30(a) shows the polar plot of the polarized IR absorbance of phenyl ring stretching observed as a function of a polarizer rotation angle in the SmA phase [92]. Large anisotropy between 0 (layer direction) and 90 (layer normal direction) is observed. In the SmX phase exhibiting V-shaped switching at zero electric ®eld, an absorption maximum of the phenyl ring stretching appears in the same direction as in SmA, as shown in Figure 9.30(b). Moreover, it is noted that the anisotropies of both phases are almost the same. This observation suggests that the molecular orientational order in the SmX phase at zero ®eld is almost the same as that in the SmA phase. By applying high dc electric ®elds in SmX to obtain the SmC* orientation, the absorption maximum direction rotates by a tilt angle of about 35 , as shown in Figure 9.30(c). If random distribution is achieved, the anisotropy must be much smaller than that in Figure 9.30(b). Actually, the absorption in the AF phase

278

H. Takezoe and Y. Takanishi

Figure 9.29. (a) The rotating angle dependences of transmittance (T±Y for F ˆ ‡6 V/mm (closed circles), 0 V/mm (open squares), and ÿ6 V/mm (open circles). (b) The apparent tilt angle yapp as a function of applied electric ®eld. (c) Dneff as a function of the applied electric ®eld. Solid and broken lines are simulated results using the collective and random models, respectively. The collective model with a layer tilt of 8 ®ts perfectly [92].

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

279

Figure 9.30. Polar plots of the absorbances of the phenyl ring (1605 cmÿ1 ) stretching mode versus the polarizer rotation angle in a homogeneously aligned liquid crystal cell: (a) SmA; (b) SmX* without applied ®eld; (c) SmX* with F ˆ ‡5 V/mm; and (d) antiferroelectric without applied ®eld [92].

…20  C† using the same cell shows small anisotropy at zero ®eld, as shown in Figure 9.30(d), because of the antiferroelectric molecular ordering. These results clearly show that the SmX* phase exhibiting V-shaped switching de®nitely has a di¨erent molecular orientation from that in the antiferroelectric phase and the random orientation but has rather uniform orientation. Second-harmonic generation (SHG) also gives useful information. Strong optical SHG signals have been observed unexpectedly at normal incidence from the V-shaped switching liquid crystal cell [92], [93]. This result seemed to be also explained by the two-dimensional Langevin potential, supporting the random model [93]. However, the collective model is much more appropriate to interpret the V-shaped switching than the random model, as will be shown in the following.

280

H. Takezoe and Y. Takanishi

Figure 9.31. Second-harmonic generation intensity pro®les as a function of the applied electric ®eld for four input/output polarization combinations, i.e., p±p ( p-in/ p-out), s±p, p±s, and s±s at normal incidence in the optical geometry shown at the top of the ®gure [94].

Figure 9.31 shows the SHG intensity pro®les as a function of an applied electric ®eld for four input/output polarization combinations, i.e., p±p ( p-in/ p-out), s±p, p±s, and s±s at normal incidence [94], where p- and s-polarizations are de®ned as polarizations parallel and perpendicualr to the smectic layer. These SHG behaviors are characteristics of the V-shaped switching because the SHG cannot be observed in normal ferroelectric bistable switching, in which the dipole direction in the ®eld-induced ferroelectric uniform state is perpendicular to the electric ®eld direction of incident light and switches instantaneously to the opposite direction. Solid curves in Figure 9.31 show the best ®tted SHG intensity pro®les simulated using the collective model. As shown in the ®gure, it is clear that the calculated SHG pro®les are identically the same as the experimental results.

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

9.5.3

281

Mechanism of V-Shaped Switching

One of the major advantages of using SHG measurement can be attributed to SHG interferometry, which gives the information of the phase of the signal or the orientational sense of the polar order. For the SHG interference measurement, a crystal quartz plate was inserted in the optical path [95]. Then, second-harmonic waves were generated both from the quartz plate and the sample and interfered with each other. A fused silica plate was located between the quartz plate and the sample cell, and was rotated about the axis parallel to the fused silica plate and perpendicular to the optical path, producing a relative phase change between second-harmonic waves from the two sources because of a frequency dispersion of the refractive index of the fused silica plate. In this way, one can observe interference fringes. Figure 9.32 shows interferograms of SHG peaks obtained by rotating the fused silica plate for positive and negative slopes of applied ®elds during the switching [94]. As shown in Figure 9.32(a), it is obvious that the phase of the SHG fringe is not reversed in the opposite slopes of an applied electric ®eld for p±p geometry. This fact indicates that the (nonlinear) polarization at about zero ®eld has the same orientational sense in both the positive and negative slopes of a ®eld. Namely the polarization switches on one-half of the smectic cone back and forth. Note that, for p±s, the phase of interferogram for inner SHG peaks (Figure 9.32(b)) is out of phase to that for outer SHG peaks (Figure 9.32(c)). Now the question is why and how molecules choose one-half of the cone for the switching. If the smectic layer is of bookshelf type, there is no reason to choose either half of the cone. However, there exists a chevron structure, as was con®rmed using a separate cell by X-ray analysis. Then molecules can choose one-half (actually less than 180 ) of the azimuthal angle on the cone, since molecules have a tendency to align themselves parallel to a substrate surface, as illustrated in Figure 9.33 (E ˆ 0). If the volume of the upper and lower halves of the chevron structure is not the same, SHG would be observed. This situation is illustrated in Figure 9.33: The liquid crystal molecules rotate toward the opposite directions in the upper and lower halves of the chevron structure. In this way, collective molecular switching occurs, as illustrated in Figure 9.33. It is clear that the directional senses of the p-polarized second-harmonic light in positive and negative slopes of the ®eld are the same, as experimentally observed. On the other hand, it is also easy to understand that two peaks appear for the s-polarized SHG at positive and negative sides of the zero ®eld and have the opposite phases, and that the outer (inner) peaks in positive and negative slopes have the same phase, as also observed experimentally. Thus, the simple collective model shown in Figure 9.33 satis®es all the experimental results of SHG intensity and SHG interferometry. The next problem is the cause and the mechanism of the V-shaped

282

H. Takezoe and Y. Takanishi Figure 9.32. Interference fringes of (a) SHG peak for p±p, (b) inner peak for p±s, and (c) outer peak for p±s observed by rotating a fused silica plate during positive (open circles) and negative (closed circles) slopes of an applied electric ®eld. Solid curves show the calculated interference fringes [94].

switching. The ®rst important question is why the molecules take the uniform orientation shown in Figure 9.33 at zero ®eld. We would like to point out the e¨ect of polarization space charge r given by ÿdiv P, where P is a polarization [96], [97]. It is well known that polar surface interaction stabilizes a twisted state [98], in which a splay deformation of P exists. Then the polarization space charge is produced. This e¨ect is pronounced when P becomes large. Hence molecules tend to form a uniform orientation shown in Figure 9.33 (E ˆ 0) to avoid the production of the polarization charge.

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

283

Figure 9.33. Illustration of the collective switching model for the V-shaped switching.

The twisted structure only at the surface area of less than 0.1 mm thick is allowed to give almost completely dark. The next question is why molecules switch collectively. The V-shaped switching occurs in the system where ferroelectric and antiferroelectric interactions compete and frustration between these structures takes place [99], [100]. Since such a frustrated system is very soft and the relaxation time becomes long, molecules change their steady-state orientation continuously under the surface constraint and varying ®eld, resulting in the collective motion. Questions still exist; the SmX* phase exhibiting V-shaped switching is a new phase or there are some conventional phases such as ferroelectric, antiferroelectric, and ferrielectric phases. One of the most important experiments remaining is the quantitative measurement of the tilting correlation between adjacent layers. This is a future problem.

9.6 9.6.1

Electro-optic Applications Tristable Switching

So far, four display modes have been proposed in ferroelectric and antiferroelectric display applications, as shown in Figure 9.34. A bistable switching in surface stabilized ferroelectric liquid crystals (SSFLCs) has been manufactured as a passive matrix liquid crystal display (PM-LCD). The counterpart of AFLC is a tristable switching, which is also a promising candidate for PM-LCD. In addition to these PM-LCDs, active matrix displays (AM-LCDs) are also proposed in FLC and AFLC materials, i.e., deformed helix FLCD (DHFLC) and V-shaped LCD (VLCD). In this section, PM-AFLCD and AM-VLCD will be described.

284

H. Takezoe and Y. Takanishi

Figure 9.34. Four display modes proposed in ferroelectric and antiferroelectric display applications.

Tristable switching is based on the ®eld-induced antiferroelectric±ferroelectric (AF±F) phase transition, as already mentioned in Section 9.1. A well-aligned cell is placed between crossed polarizers, one of which is parallel to the smectic layer. Then a dark view is obtained in the absence of a ®eld. By applying a ®eld, the AF±F phase transition takes place and the view turns bright due to optical birefringence. Because of this switching mode, the viewing angle is very wide. The fundamental concept of the driving scheme for PM-AFLCD [11] is shown in Figure 9.35. The driving is made by applying a pulsed voltage (VD or VN ) in addition to a bias voltage (V0 ) which is between high and low H L threshold voltages (Vth and Vth , respectively) forming a hysteresis. For H H , switching does VD ‡ V0 > Vth , switching occurs, while for VN ‡ V0 < Vth not occur. In the multiplexing, hystereses on the positive and negative sides were used alternatively. Because of this scheme, charge accumulation which causes the so-called ghost e¨ect is not so serious. The response speed depends on the e¨ective voltage that is the di¨erence between the applied voltage and the threshold voltage (Figure 9.35). Figure 9.36 shows the transmittance changes (a) from AF to F and (b) from F to AF in MHPOBC [101]. In both cases, the lower voltage of a rectangular ®eld

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

285

Figure 9.35. Fundamental concept of driving scheme for PM-AFLCD.

was ®xed at 0 V in the AF state and the voltage in the F state was varied from 32 V to 38 V. As shown in Figure 9.36(a), 28 V is too low to induce the H ). Whilst the reAF±F phase transition (below the threshold voltage Vth sponse time markedly depends on the higher voltage applied for the AF±F switching, it is independent of the higher voltage for the F±AF switching. Another noticeable feature is the two-step change for the AF±F switching. At the ®rst step, a very fast switching to a certain transmittance occurs and then slowly (particularly at low voltages) changes to the maximum transmittance in the F state. The ®rst metastable transmittance level corresponds to the transmittance in the AF state at the threshold voltage and is attributed to a pretransitional e¨ect for the AF±F transition. This phenomenon can be ascribed so that the anticlinic molecular arrangement in adjacent layers is slightly deformed owing to a change in the azimuthal angle. As a result, it leads to the leakage of the transmittance in the AF state and brings about a serious problem for display application, i.e., low contrast ratio. Therefore, the materials which exhibit negligible pretransitional e¨ect are required for application. The response time for the AF±F and F±AF switching is shown in Figure 9.37 for the stepwise voltage change (a) from 0 V and the stepwise voltage drop (b) from 30 V [101]. Figure 9.37 clearly indicates that the response time remarkably depends on the e¨ective voltage. The response time as a function

286

H. Takezoe and Y. Takanishi

Figure 9.36. The transmittance changes (a) from AF to F and (b) from F to AF in MHPOBC [101].

Figure 9.37. The response time for the AF±F and F±AF switching in MHPOBC [101].

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

287

Figure 9.38. The response time as a function of e¨ective voltage in MHPOBC [101].

Figure 9.39. 17 inch AFLC display developed by DENSO CORP.

of the e¨ective voltage is shown in Figure 9.38. The dependence of the fourth power and seventh power was obtained for the AF±F and F±AF switching, respectively. The response time is very fast, if we apply high e¨ective voltages. The applied voltage, which is equal to a sum of the e¨ective voltage and the threshold voltage, is necessarily high. This is one of the disadvantages for the tristable AFLC display. A gray level can be achieved by utilizing domain formation associated with the ®eld-induced AF±F phase transition. At the transition, ferroelectric striped domains are nucleated at the electrode edges and grow along smectic layers [101], [102]. Therefore, the area could be controled by the height or duration of the pulses applied. Actually, full color displays have been developed by several companies. Figure 9.39 shows a photograph of 17 inch AFLCs display developed by DENSO.

9.6.2

V-Shaped Switching

The V-shaped switching mode mentioned in the previous section provides us with another promising display mode. The switching is associated with nei-

288

H. Takezoe and Y. Takanishi

Figure 9.40. (a) The optical transmittance measured at 40  C by applying triangular voltage waves of 50, 100, and 1000 mHz for the cells coated with polyimide ®lms of two di¨erent thicknesses. (b) The V-shaped switching measured at a frequency of 1 Hz and at 25  C, 55  C, and 70  C for polyimide ®lms of two di¨erent thicknesses [103].

ther threshold nor hysteresis. The switching is not associated with domain formation. The cell always gives a dark appearance, whenever an applied ®eld is turned o¨. The performance is sometimes in¯uenced by surface conditions. Chandani et al. [91] systematically studied the e¨ect of the surface alignment layer using a liquid crystal compound 5 in Figure 9.26, and concluded that thick alignment layers and less polar surfaces are ideal for V-shaped switching. Figure 9.40(a) compares the optical transmittance measured at 40  C by applying triangular voltage waves of 50, 100, and 1000 mHz for the cells coated with a polyimide for DHFLC of two di¨erent thicknesses [103]. At these frequencies, V-shaped and W-shaped switchings were favorable for Ê and 600 A Ê ®lms, respectively. The bottom line (abscissa) corresponds 1360 A to the complete dark state when no light goes to the photomultiplier tube. A comparison of the V-shaped switching measured at a frequency of 1 Hz with varied temperatures at 25  C, 55  C, and 70  C for two di¨erent PI-DHFLC ®lm thicknesses is shown in Figure 9.40(b). The solid straight line in each ®gure shows the transmittance when the ®eld is switched o¨. For both thicknesses, the transmittance remains at low level when the ®eld is switched o¨, unless the temperature is high (70  C) and the PI-DHFLC ®lm is thin Ê ). Thus a thicker alignment layer assists in achieving V-shaped (600 A switching [103]. However, when the alignment layer becomes thicker, a higher driving ®eld is necessary; this may be a disadvantage from an application point of view. Instead of using a thick polyimide ®lm, the capacitance was decreased by Ê ) together with an insulating layer, i.e., using a thin polyimide ®lm (600 A

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

289

Figure 9.41. (a) The optical transmittance measured at 40  C and at di¨erent Ê ) with and without the insulating layer. (b) frequencies for polyimide SP550 (600 A The temperature-dependence of the optical response measured at 1 Hz frequency for the cells with and without the insulating layer [103].

Ê thickness. Figure 9.41(a) compares the optical transmittance TaOX of 900 A measured at 40  C and at di¨erent frequencies for a polyimide ®lm (Toray, Ê ) with and without the insulating layer. As the SP550 ®lm SP550, 600 A Ê ) it gives rise to W-shaped switching without an is relatively thinner (600 A insulating layer. When TaOX was used as the insulating layer with a thin SP550, good V-shaped switching was observed. Figure 9.41(b) illustrates the temperature-dependence of the optical response measured at 1 Hz frequency for the cells with and without the insulating layer. The transmittance in the ®eld-o¨ state became very low when the insulating layer was used. From this ®gure it is clear that good V-shaped switching can be achieved when TaOX is used as an insulating layer together with thin PI-DHFLC and SP550 at all temperatures and frequencies. Thus, less polar TaOX , which has a high dielectric constant, favors V-shaped switching. In this way, to achieve an ideal V-shaped switching, an appropriate choice of a surface alingment layer and an insulating layer in addition to smectic liquid crystal materials is very important. Because of no threshold behavior, the mode can be utilized as AM-LCD in conjunction with TFT. The continuous change in the transmittance is suitable for full color display. Since a very dark view is realized in the absence of a ®eld, a high contrast ratio more than 100 could be easily achieved. At present at least Toshiba and Casio announced their prototype LCD using V-shaped switching, as shown in Figure 9.42. Speci®cations of these displays in addition to DENSO PM-AFLCD are shown in Figure 9.43. These photographs clearly reveal a good viewing angle. Because of its high display performance such as a wide viewing angle, high contrast, full color, and high

290

H. Takezoe and Y. Takanishi

Figure 9.42. AM-VLCD developed by Toshiba and Casio.

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

291

Figure 9.43. Speci®cations of the display by DENSO CORP., Toshiba and Casio.

speed, the V-shaped switching display is considered as a candidate for nextgeneration display.

References [1] R.B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, J. Phys. (France) 36, L69 (1975). [2] T. Inukai, K. Furukawa, K. Terashima, S. Saito, M. Isogai, T. Kitamura, and A. Mukoh, Abstract Book of Japan Domestic Liq. Cryst. Meeting, Kanazawa, p. 172 (in Japanese), 1985. [3] N. Hiji, A.D.L. Chandani, S. Nishiyama, Y. Ouchi, H. Takezoe, and A. Fukuda, Ferroelectrics 85, 99 (1988). [4] K. Furukawa, K. Terashima, M. Ichihashi, S. Saitoh, K. Miyazawa, and T. Inukai, Ferroelectrics 85, 63 (1988). [5] A.M. Levelut, C. Germain, P. Keller, and L. Liebert, J. Phys. (France) 44, 623 (1983). [6] Y. Galerne and L. Liebert, Abstract Book of 2nd International Conf. Ferroelectric Liq. Cryst. O27 (1989). [7] H. Takezoe, A.D.L. Chandani, J. Lee, E. Gorecka, Y. Ouchi, A. Fukuda, K. Terashima, K. Furukawa, and A. Kishi, Abstract Book of 2nd International Conf. Ferroelectric Liq. Cryst. O26 (1989). [8] H. Takezoe, A.D.L. Chandani, E. Gorecka, Y. Ouchi, and A. Fukuda, Abstract Book of 2nd International Conf. Ferroelectric Liq. Cryst. P108 (1989). [9] J.W. Goodby and E. Chin, Liq. Cryst. 3, 1245 (1988). [10] A.D.L. Chandani, T. Hagiwara, Y. Suzuki, Y. Ouchi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 27, L729 (1988).

292

H. Takezoe and Y. Takanishi

[11] N. Yamamoto, Y. Yamada, N. Koshobu, K. Mori, K. Nakamura, H. Orihara, Y. Ishibashi, Y. Suzuki, and I. Kawamura, Jpn. J. Appl. Phys. 31, 3182 (1992). [12] A.D.L. Chandani, E. Gorecka, Y. Ouchi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 28, L1265 (1989). [13] Y. Galerne and L. Liebert, Phys. Rev. Lett. 64, 906 (1990). [14] Y. Takanishi, H. Takezoe, M. Johno, T. Yui, and A. Fukuda, Jpn. J. Appl. Phys. 32, 4605 (1993). [15] P. Cladis and H.R. Brand, Liq. Cryst. 14, 1327 (1993). [16] G. Heppke, P. Kleineberg, D. LoÈtzsch, S. Mery, and R. Shashidhar, Mol. Cryst. Liq. Cryst. 231, 257 (1993). [17] Ch. Bahr and D. Fliegner, Phys. Rev. Lett. 70, 1842 (1993). [18] Y. Takanishi, H. Takezoe, A. Fukuda, H. Komura, and J. Watanabe, J. Mater. Chem. 2, 71 (1992) [19] Y. Takanishi, H. Takezoe, A. Fukuda, and J. Watanabe, Phys. Rev. B 45, 7684 (1992). [20] E. Gorecka, A.D.L. Chandani, Y. Ouchi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 29, 131 (1990). [21] T. Fujikawa, K. Hiraoka, T. Isozaki, K. Kajikawa, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 32, 985 (1993). [22] K. Hiraoka, A. Taguchi, Y. Ouchi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 29, L103 (1990). [23] K. Hiraoka, Y. Ouchi, H. Takezoe, and A. Fukuda, Mol. Cryst. Liq. Cryst. 199, 197 (1991). [24] K. Itoh, Y. Takanishi, J. Yokoyama, K. Ishikawa, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 36, L784 (1997). [25] Y. Suzuki, T. Hagiwara, I. Kawamura, N. Okamura, T. Kitazume, M. Kakimoto, Y. Imai, Y. Ouchi, H. Takezoe, and A. Fukuda, Liq. Cryst. 6, 167 (1989). [26] H. Takezoe, J. Lee, A.D.L. Chandani, E. Gorecka, Y. Ouchi, and A. Fukuda, Ferroelectrics 114, 187 (1991). [27] T. Isozaki, H. Takezoe, A. Fukuda, Y. Suzuki, and I. Kawamura, J. Mater. Chem. 4, 237 (1994). [28] H. Takezoe, A. Fukuda, A. Ikeda, Y. Takanishi, T. Umemoto, J. Watanabe, H. Iwane, M. Hara, and K. Itoh, Ferroelectrics 122, 167 (1991). [29] A. Ikeda, Y. Takanishi, H. Takezoe, A. Fukuda, S. Inui, S. Kawano, M. Saito, and H. Iwane, Jpn. J. Appl. Phys. 30, L1032 (1991). [30] S. Inui, S. Kawano, M. Saito, H. Iwane, Y. Takanishi, K. Hiraoka, Y. Ouchi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 30, L1032 (1990). [31] A. Ikeda, Y. Takanishi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 32, L97 (1993). [32] Y. Takanishi, K. Hiraoka, V.K. Agrawal, H. Takezoe, A. Fukuda, and M. Matsushita, Jpn. J. Appl. Phys. 30, 2023 (1991). [33] K. Miyachi, J. Matsushima, Y. Takanishi, K. Ishikawa, H. Takezoe, and A. Fukuda, Phys. Rev. E 52, R2158 (1995). [34] I. Nishiyama and J.W. Goodby, J. Mater. Chem. 3, 149 (1993). [35] T. Kusumoto, T. Isozaki, Y. Suzuki, Y. Takanishi, H. Takezoe, A. Fukuda, and T. Hiyama, Jpn. J. Appl. Phys. 34, L830 (1995). [36] Y. Suzuki, T. Isozaki, S. Hashimoto, T. Kusumoto, T. Hiyama, Y. Takanishi, H. Takezoe, and A. Fukuda, J. Mater. Chem. 6, 753 (1996).

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69]

293

J. Watanabe and M. Hayashi, Macromolecules 22, 4083 (1989). K. Hori and K. Endo, Bull. Chem. Soc. Jpn. 66, 46 (1993). K. Hori, S. Kawahara, and K. Ito, Ferroelectrics 147, 91 (1993). S. Yoshida, B. Jin, Y. Takanishi, K. Tokumaru, K. Ishikawa, H. Takezoe, A. Fukuda, T. Kusumoto, T. Nakai, and S. Miyajima, J. Phys. Soc. Jpn. 68, 9 (1999). T. Nakai, S. Miyajima, Y. Takanishi, S. Yoshida, and A. Fukuda, J. Phys. Chem. B 103, 406 (1999). B. Jin, Z. Ling, Y. Takanishi, K. Ishikawa, H. Takezoe, A. Fukuda, M. Kakimoto, and T. Kitazume, Phys. Rev. E 53, R4295 (1996). Y. Ouchi, Y. Yoshioka, H. Ishii, K. Seki, M. Kitamura, R. Noyori, Y. Takanishi, and I. Nishiyama, J. Mater. Chem. 5, 2297 (1995). N. Okabe, Y. Suzuki, I. Kawamura, T. Isozaki, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 31, L793 (1992). A.D.L. Chandani, Y. Ouchi, H. Takezoe, A. Fukuda, K. Terashima, K. Furukawa, and A. Kishi, Jpn. J. Appl. Phys. 28, L1261 (1989). M. Fukui, H. Orihara, Y. Yamada, N. Yamamoto, and Y. Ishibashi, Jpn. J. Appl. Phys. 28, L849 (1989). K. Ema, H. Yao, I. Kawamura, T. Chan, and C.W. Garland, Phys. Rev. E 47, 1203 (1993). S. Asahina, M. Sorai, A. Fukuda, H. Takezoe, K. Furukawa, K. Terashima, Y. Suzuki, and I. Kawamura, Liq. Cryst. 23, 339 (1997). K. Itoh, M. Kabe, K. Miyachi, Y. Takanishi, K. Ishikawa, H. Takezoe, and A. Fukuda, J. Mater. Chem. 7, 407 (1997). T. Isozaki, T. Fujikawa, H. Takezoe, A. Fukuda, T. Hagiwara, Y. Suzuki, and I. Kawamura, Jpn. J. Appl. Phys. 31, L1435 (1992). T. Isozaki, T. Fujikawa, H. Takezoe, A. Fukuda, T. Hagiwara, Y. Suzuki, and I. Kawamura, Phys. Rev. B 48, 13439 (1993). T. Isozaki, H. Takezoe, A. Fukuda, Y. Suzuki, and I. Kawamura, J. Mater. Chem. 4, 237 (1994). T. Isozaki, K. Ishikawa, H. Takezoe, and A. Fukuda, Ferroelectrics 147, 121 (1993). A. Fukuda, Y. Takanishi, T. Isozaki, K. Ishikawa, and H. Takezoe, J. Mater. Chem. 4, 997 (1994). P. Bak and R. Bruinsma, Phys. Rev. Lett. 49, 249 (1982). R. Bruinsma and P. Bak, Phys. Rev. B 27, 5824 (1983). P. Bak, Physics Today 39, 38 (1986). M. Yamashita and S. Miyajima, Ferroelectrics 148, 1 (1993). M. Yamashita, Mol. Cryst. Liq. Cryst. 263, 93 (1995). M. Yamashita, Ferroelectrics 181, 201 (1996). M. Yamashita, J. Phys. Soc. Jpn. 65, 2122 (1996). M. Yamashita, J. Phys. Soc. Jpn. 65, 2904 (1996). T. Koda and H. Kimura, J. Phys. Soc. Jpn. 64, 3787 (1995). H. Orihara and Y. Ishibashi, Jpn. J. Appl. Phys. 29, L115 (1990). H. Sun, H. Orihara, and Y. Ishibashi, J. Phys. Soc. Jpn. 62, 2706 (1993). B. Zeks, R. Blinc, and M. Cepic, Ferroelectrics 122, 221 (1991). B. Zeks and M. Cepic, Liq. Cryst. 14, 445 (1993). V.L. Lorman, A.A. Bulbitch, and P. Toledano, Phys. Rev. E 49, 1367 (1994). M. Cepic and B. Zeks, Mol. Cryst. Liq. Cryst. 263, 61 (1995).

294 [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96]

H. Takezoe and Y. Takanishi A. Roy and N.V. Madhusudana, Europhys. Lett. 36, 221 (1996). V.L. Lorman, Mol. Cryst. Liq. Cryst. 262, 437 (1995). A. Roy and N.V. Madhusudana, Europhys. Lett. 41, 501 (1998). P. Mach, R. Pindak, A.-M. Levelut, P. Barois, H.T. Nguyen, C.C. Hunag, and L. Furenlid, Phys. Rev. Lett. 81, 1015 (1998). J. Li, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 30, 532 (1991). K. Miyachi, M. Kabe, K. Ishikawa, H. Takezoe, and A. Fukuda, Ferroelectrics 147, 147 (1993). T. Akizuki, K. Miyachi, Y. Takanishi, K. Ishikawa, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 38, 4832 (1999). T. Isozaki, K. Hiraoka, Y. Takanishi, H. Takezoe, A. Fukuda, Y. Suzuki, and I. Kawamura, Liq. Cryst. 12, 59 (1992). M. Skarabot, M. Cepic, B. Zeks, R. Blinc, G. Heppke, A.V. Kityk, and I. Musevic, Phys. Rev. E 58, 575 (1998). K. Hiraoka, Y. Takanishi, K. Skarp, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 30, L1819 (1991). C. Bahr, D. Fliegner, C.J. Booth, and J.W. Goodby, Phys. Rev. E 51, 3823 (1995). V. Laux, N. Isaert, H.T. Nguyen, P. Cluseau, and C. Destrade, Ferroelectrics 179, 25 (1996). K. Yamada, Y. Takanishi, K. Ishikawa, H. Takezoe, A. Fukuda, and M.A. Osipov, Phys. Rev. E 56, R43 (1997). A. Rastegar, M. Ochsenbein, I. Musevic, T. Rasing, and G. Heppke, Ferroelectrics 212, 249 (1998). K. Ema, M. Kanai, H. Yao, Y. Takanishi, and H. Takezoe, Phys. Rev. E 61, 1585 (2000). S. Inui, N. Iimura, T. Suzuki, H. Iwane, K. Miyachi, Y. Takanishi, and A. Fukuda, J. Mater. Chem. 6, 71 (1996). A. Fukuda, S.S. Seomun, T. Takahashi, Y. Takanishi, and K. Ishikawa, Mol. Cryst. Liq. Cryst. 303, 379 (1997). C. Tanaka, T. Fujiyama, T. Maruyama, and S. Nishiyama, Abst. of 21st Jpn. Liq. Cryst. Conf. (Sendai), 2C18 (1995). S.S. Seomun, Y. Takanishi, K. Ishikawa, H. Takezoe, A. Fukuda, C. Tanaka, T. Fujiyama, T. Maruyama, and S. Nishiyama, Mol. Cryst. Liq. Cryst. 303, 181 (1997). S.S. Seomun, Y. Takanishi, K. Ishikawa, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 36, 3586 (1997). S.S. Seomun, T. Gouda, Y. Takanishi, K. Ishikawa, H. Takezoe, and A. Fukuda, Liq. Cryst. 26, 151 (1999). A.D.L. Chandani, Y. Cui, S.S. Seomun, Y. Takanishi, K. Ishikawa, H. Takezoe, and A. Fukuda, Liq. Cryst. 26, 167 (1999). B. Park, S.S. Seomun, M. Nakata, M. Takahashi, Y. Takanishi, K. Ishikawa, and H. Takezoe, Jpn. J. Appl. Phys. 38, 1474 (1999). S.S. Seomun, B. Park, A.D.L. Chandani, D.S. Hermann, Y. Takanishi, K. Ishikawa, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 37, L691 (1998). B. Park, M. Nakata, S.S. Seomun, Y. Takanishi, K. Ishikawa, and H. Takezoe, Phys. Rev. E 59, R3815 (1999). S.-W. Choi, Y. Kinoshita, B. Park, H. Takezoe, T. Niori, and J. Watanabe, Jpn. J. Appl. Phys. 37, 3408 (1998). K. Okano, Jpn. J. Appl. Phys. 25, L846 (1986).

9. Smectic Liquid Crystals: Antiferroelectric and Ferrielectric Phases

295

[97] M. Nakagawa and T. Akahane, J. Phys. Soc. Jpn. 55, 1516 (1986). [98] Y. Ouchi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 26, 1 (1987). [99] E. Gorecka, D. Pociecha, M. Glogarova, and J. Mieczkowski, Phys. Rev. Lett. 81, 2946 (1998). [100] S. Tanaka and M. Yamashita, Jpn. J. Appl. Phys. 38, L139 (1999). [101] M. Johno, K. Itoh, J. Lee, Y. Ouchi, H. Takezoe, A. Fukuda, and T. Kitazume, Jpn. J. Appl. Phys. 29, L107 (1990). [102] Y. Yamada, K. Mori, N. Yamamoto, H. Hayashi, K. Nakamura, M. Yamawaki, H. Orihara, and Y. Ishibashi, Jpn. J. Appl. Phys. 28, L1606 (1989). [103] A.D.L. Chandani, Y. Cui, S.S. Seomun, Y. Takanishi, K. Ishikawa, H. Takezoe, and A. Fukuda, Mol. Cryst. Liq. Cryst. 322, 337 (1998).

As for the molecular orientation structures of the SmCg* and AF phases discussed on p. 273, recent detailed ellipsometry measurements succeeded in modeling the distorted structures that are distorted from the Ising model and clock model (P.M. Johnson, D.A. Olson, S. Pankratz, T. Nguyen, J. Goodby, M. Hird, and C.C. Huang, Phys. Rev. Lett. 84, 4870 (2000)) and are essentially the same as a distorted Ising model [76].

10

Twist Grain Boundary Phases Heinz-Siegfried Kitzerow

Twist grain boundary (TGB) phases have been known since 1988 and have attracted great attention during the last 10 years. Chiral liquid crystals have the tendency to form a cholesteric-like helical director ®eld. On the other hand, the molecular interactions may favor a smectic layer structure. However, it is impossible to realize a continuous structure which exhibits both a cholesteric director ®eld and a smectic layer structure at the same time. The competition between these two structural features can result in frustrated structures containing a regular lattice of grain boundaries which in turn consist of a lattice of screw dislocations. This defect structure exhibits an interesting theoretical analogy to the ¯ux line lattice which occurs in the Shubnikov phase of type II superconductors. However, the range of parameters determining the structure is larger in liquid crystals than in superconductors. Thus, a large variety of new phases, such as the TGBA , TGBC , TGBC  , TGB2q , melted grain boundary (MGB) phases, a defect line liquid NL , antiferroelectric crystals of twist grain boundaries, and smectic blue phases have been predicted and/or experimentally observed. The extensive studies on defect structures and phase transitions make this subject relevant and exciting to both experts and non-specialists. A review of experiments is given in this chapter.

10.1

Introduction

Twist grain boundary (TGB) phases [1]±[4] usually appear in the temperature range between a cholesteric N* phase with short pitch and a smectic phase, typically SmA or SmC*. In particular, they are expected to appear close to a N*/SmA/SmC* triple point [5]. One of their remarkable properties is the selective re¯ection of circularly polarized light [2], [3]. This feature shows that the director ®eld has a helical structure similar to the cholesteric phase. On the other hand, X-ray investigations of TGB phases indicate a layer structure as occurring in smectic phases [6]. Chirality of the system is an essential precondition for the occurrence of TGB phases. In mixtures of 296

10. Twist Grain Boundary Phases

297

a chiral and nonchiral component (e.g., chiral±racemic mixtures), the concentration of the chiral component (or the enantiomeric excess) must exceed a critical value in order to lead to TGB structures [7]. Figure 10.1 shows the structure of the TGBA phase which consists of smectic slabs, separated by defect walls. Neighboring slabs are tilted with respect to each other by an angle Dw0 , thereby forming a helical structure. The order within each slab is equivalent to the smectic-A (SmA) phase: The molecules are uniaxially aligned along the local director n and arranged in layers. The layer normal Q is parallel to n. The layer spacing d ˆ 2p=jQj is in the range of a few nm. The slabs have a typical thickness lb of some 10 nm. The grain boundaries between the slabs are defect walls consisting of parallel defect lines (twist dislocations). The distance ld between the defect lines is similar to the slab thickness lb in TGBA [1], [6], [8] (however, in TGBC : lb > ld ). The director n forms a helical structure due to the tilt between neighboring slabs. The helix axis h is perpendicular to the director and the pitch p is in the range from a few hundred nm to a few mm. Geometrical considerations lead to the following relations between the tilt angle Dw0 and the characteristic lengths sin…Dw0 =2† ˆ d=…2ld †; a :ˆ Dw0 =…2p† ˆ lb = p ˆ

2 sinÿ1 …12 d=ld †=2p A d=…2pld †:

…10:1† …10:2†

The parameter a can be rational (i.e., a ˆ m=n, where m and n are integers) or irrational [1]. If a is irrational, the structure is incommensurate, i.e., there is no periodicity of the orientation of the slabs along the pitch axis. If a is rational, the system is commensurate and has a n-fold screw axis. The TGB phase has a periodic crystalline symmetry if the screw axis is crystallographically allowed (n ˆ 2; 3; 4, or 6), but a quasi-crystalline structure if n 0 2; 3; 4 or 6 [1], [9]. The peculiar structure of the TGBA phase can be considered as a compromise between the incompatible properties of the cholesteric phase which appears at higher temperatures and the SmA phase which appears at lower temperatures. The cholesteric phase is characterized by a helical director ®eld and shows no positional order of the molecules. Thus, its director ®eld can be deformed due to surface e¨ects or external ®elds. The increase of the Gibbs free energy due to the elastic deformations of nematic and cholesteric phases is given by … …10:3† Gela ˆ G0 ‡ d 3 rfgela …r†g; where G0 is the Gibbs free energy of the undeformed state and the free energy density gela …r† is given by [6]: gela …r† ˆ 12 K11 …`
n† 2 ‡ 12 K22 …n
`  n† 2 ‡ 12 K33 ‰n  …`  n†Š 2 ‡ h…n
`  n†

298

H.-S. Kitzerow

(a)

(b) Figure 10.1. (a) Structure of a phase which shows a local smectic order and a helical director ®eld at the same time, as proposed by Pollmann [21] and MuÈller (from [20]). (b) Structure of the TGBA phase proposed by Renn and Lubensky. In addition to the structural features displayed in (a), the grain boundaries and the screw dislocations are shown. Characteristic lengths: d ˆ smectic layer spacing, lb ˆ thickness of the smectic slabs, ld ˆ distance between neighboring screw dislocations, and p ˆ pitch of the director ®eld (reprinted from [125]). (c) Smectic layers of two neighboring blocks above (solid lines) and below the drawing plane (dashed lines), and screw dislocations (vertical tubes) between the two blocks. The relation between the layer spacing d and the distance of the defect lines ld is given by (10.1).

10. Twist Grain Boundary Phases

299

(c) Figure 10.1 (continued )

ˆ 12 fK11 …`
n† 2 ‡ K22 ‰…n
`  n† ‡ q0 Š 2 ‡ K33 ‰n  …`  n†Š 2 ÿ K22 q02 g;

…10:4†

where the ®rst three terms correspond to splay, twist, and bend deformation, respectively (Figure 10.2). The elastic coe½cients K11 , K22 , and K33 [10]±[12] are of the order 10ÿ12 ±10ÿ11 N [13]. The coe½cient h describes the chirality of the system. A nonchiral nematic phase is characterized by q0 ˆ h=K22 ˆ 0. The intrinsic twist of the cholesteric phase is described by the pitch p0 ˆ 2p=jq0 j ˆ 2pK22 =h, for example, an undisturbed right-handed cholesteric structure with a pitch axis q ˆ …q; 0; 0† along the x-direction of Cartesian coordinates is described by n…r† ˆ …0; cos…q
r†; sin…q
r††, and thus n
…`  n† ˆ ÿq, so that the twist energy is minimized by jqj ˆ q0 . In contrast to the cholesteric phase, the SmA phase shows a layer structure which is incompatible with any director ®eld n…r† which is twisted or bent.1 The following consideration shows that terms containing `  n cannot occur in „ smectic liquid crystals [11], [14]: The value of the path integral d ÿ1 n
dr along a path within a SmA structure corresponds to the number of layers which are† passed along the way. Thus, its value should be independent on the path, n
dr ˆ 0. For a continuous structure, this condition is equivalent to `  n ˆ 0, which means that the twist and bend term of (10.4) cannot occur in the SmA phase. Indeed, experiments have shown that the pitch p0 and the elastic coe½cients K22 and K33 of the cholesteric phase increase strongly with decreasing temperature in the vicinity of a SmA phases and diverge at the N*/SmA transition [15], [16]. The only way to twist or bend

1 In analogy, the reader may try to twist a pile of paper sheets.

300

H.-S. Kitzerow

Figure 10.2. (a)±(c) Basic deformation modes of a nematic director ®eld: (a) splay deformation …div n 0 0†; (b) twist deformation …n
curl n 0 0†; and (c) bend deformation …n  curl n 0 0†. (d)±(f ) The same deformations of the director ®eld in a smectic phase. Only the splay deformation of the director ®eld (d) is compatible with the constant layer spacing. A twist deformation (e) [bend deformation (f )] is only possible if screw dislocations [edge dislocations] appear.

the director ®eld of a layer structure is to insert twist- or edge-dislocations, respectively (Figure 10.2).2 In 1972, de Gennes [14], [17] pointed out that there is a phenomenological analogy between the smectic/nematic phase transition in liquid crystals and the superconducting/normal-conducting transition in metals. Due to this analogy, he predicted that a regular defect structure should occur in smectic liquid crystals close to the smectic/nematic transition when a twist or bend deformation is exerted on the director ®eld of the smectic phase. This defect structure is the analog of the mixed state in type II superconductors [18] which exhibits a lattice of magnetic ¯ux lines [19]. For many years, the prediction of corresponding liquid crystal structures seemed not to be con®rmed by experiments. Nevertheless, chiral systems showed eventually a peculiar behavior at the SmA/N* transition, for example, MuÈller [20] obtained two 2 In order to test this, try twisting the pile of paper after a cut was made from the center to the edge of each sheet.

10. Twist Grain Boundary Phases

301

di¨erent SmA/N* transition temperatures due to dilatometric and optical studies. Pollmann proposed the intermediate state given in Figure 10.1(a) [21]. In 1988, Renn and Lubensky [1] extended the theory of the smectic/ nematic transition to chiral systems and pointed out that the intrinsic twist of the cholesteric phase can act as an external ``®eld,'' just like the mechanical deformation considered by de Gennes [14], thereby leading to TGB phases in chiral systems. At the same time, and independent of the theoretical prediction, Goodby et al. [2], [3] synthesized 14P1M7, the ®rst compound which was used to obtain experimental evidence for the existence of a TGBA phase. Since then, a variety of further phases consisting of smectic slabs have been studied: In TGBC phases [22]±[25], the director is tilted with respect to the layer normal. The TGBC  phase [26]±[28] even shows a twisted director ®eld within each smectic slab. The following sections give brief summaries of the theory (10.2) and the chemical compounds showing TGB phases (10.3). Subsequently, the properties of di¨erent TGB phases are reviewed, including TGB2q [29], a twisted line liquid NL [30], [31], smectic blue phases [32], and smectic-Q [33]. More details can be found in some excellent previous reviews which are focused on di¨erent aspects [34]±[39]. For completeness, it is necessary to mention that in the early experimental papers, the TGBA phase was denoted as ``SmA*.'' However, today, some authors denote the usual untwisted smectic-A phase as ``SmA*'' if it consists of chiral molecules. The purpose of using this notation for the untwisted smectic-A phase is to express that a smectic-A phase can own particular properties, such as a non-vanishing electroclinic coe½cient, due to the chirality of its molecules, although the arrangement of the molecules is not twisted. Due to this ambiguity, the abbreviation TGB with the respective subscript is used for twist grain boundary phases throughout in order to avoid any confusion with the dislocation-free smectic-A phase. (Very few authors [28], [40], [41] use the additional abbreviations Aa for TGBA , Ca for TGBC , and Ca* for TGBC  .)

10.2

The Analogy Between TGB Phases and Superconductors

In the temperature range close to the transition temperature TSmA ±N , the phase transition from the smectic A to the nematic phase can phenomenologically be described by the Landau±de Gennes theory [12], [42], [43]. The free energy density is expanded in a power series of an order parameter c: gL ˆ 12 fajcj 2 ‡ 12 bjcj 4 ‡ 13 gjcj 6 g;

…10:5†

where a ˆ a…T ÿ TSmA ±N †, and a > 0. Depending on the values of b and g, a transition of ®rst order …b < 0; g > 0†, tricritical behavior …b ˆ 0; g > 0†, or a transition of second order …b > 0; g ˆ 0† can be described (Figure 10.3). The order parameter c is nonzero in the low temperature phase (smectic phase) and vanishes in the high-temperature phase (nematic phase). The

302

H.-S. Kitzerow

Figure 10.3. Free-energy density as a function of the order parameter for di¨erent temperatures (top) and the corresponding relation between the temperature and the order parameter (bottom) in the vicinity of (a) a second-order and (b) a ®rst-order phase transition.

layer structure of smectic phases is characterized by a periodic density r…r† ˆ r0 ‡ r1 cos…Q
r ‡ F…r†† ˆ r0 ‡ 12 r1 ‰exp……iQ
r ‡ F…r†† ‡ c:c:Š;

…10:6†

where Q is a layer normal with jQj ˆ 2p=d, and d is the layer spacing. The density of a nematic phase is spatially uniform. Thus, the deviation from the average density may be written as [1]: r…r† ÿ r0 ˆ: c…r† ‡ c …r†

…10:7†

and c…r† can be choosen as an appropriate order parameter to describe the SmA±N transition. (Alternatively, the complex coe½cient r1 exp…F…r††

10. Twist Grain Boundary Phases

303

containing only the amplitude and the phase of the density modulation can serve as an order parameter [12].) Since the director and the smectic order parameter depend on the position r, the expression of (10.5) has to be completed by a gradient term gG which corresponds to displacements of the smectic layers in the SmA phase, and by the Frank elastic energy (10.4). If the layer normal and the director are oriented along the z-direction, the gradient term reads [1]: gG ˆ 12 f‰…Ck ÿ C? †ni nj ‡ C? dij Š‰…` ÿ iQ0 n†i c…` ‡ iQ0 n†j c Šg: …10:8† If the amplitude of the complex order parameter c is constant, (10.8) describes changes of the free energy which are due to compression or dilation of the layers (i.e., deviations of jQj from jQ0 j) or due to phase shifts of c, i.e., displacements of the smectic layers. In contrast to the gradient term appearing in the Ginzburg±Landau ansatz [44], two coe½cients, Ck and C? , appear because of the anisotropy of the liquid crystal. In a one-constant approximation, Ck ˆ C? ˆ: C, (10.8) is reduced to gG ˆ 12 fC? j…` ÿ iQ0 n†cj 2 g:

…10:9†

This term describes the coupling between the director and the smectic order parameter. It vanishes if c…r† z exp…iQ
r† with jQj ˆ jQ0 j and Qkn, because the layers are perpendicular to the director in the SmA phase. However, any deviation dn of the director n from its equilibrium value n0 causes a rise of the free energy (Figure 10.4). If the director and the layer normal are oriented along the z-direction, a small deviation dnx of the director along the x-direction is energetically equivalent to a displacement u of the layers along the z-direction by approximately u ˆ dnx x which in turn corresponds to a phase shift F…r† ˆ ÿQ0 dnx x. The corresponding contribution to the free-energy density is CQ02 dnx2 jcj 2 if the layers are unchanged. However, a displacement of the layers by u ˆ ÿdnx x, corresponding to a phase shift of F…r† ˆ ‡Q0 dnx x can compensate the director reorientation. Combining (10.4), (10.5), and (10.9) leads to the free energy [1]: g ˆ g0 ‡ 12 fajcj 2 ‡ 12 bjcj 4 ‡ Cj…` ÿ iQn†cj 2 g ‡ 12 fK11 …`
n† 2 ‡ K22 ‰…n
`  n† ‡ q0 Š 2 ‡ K33 ‰n  …`  n†Š 2 ÿ K22 q02 g

…10:10†

for a second-order transition …g ˆ 0†3. This expression is very similar to the

3 Using the complex amplitude of the densitiy variation C :ˆ r1 exp…F…r†† as an order parameter leads to the free energy g ˆ g0 ‡ 12 fajCj 2 ‡ 12 bjCj 4 ‡ Ck jqC=qzj 2 ‡ C? j…`? ÿ iQ dn? †Cj 2 g ‡ gela ; where the local z-axis is chosen perpendicular to the smectic layers, dn? :ˆ n ÿ z is the local deviation of the director from the layer normal, and `? :ˆ …q=qx; q=q y† [12], [37].

304

H.-S. Kitzerow

Figure 10.4. Coupling between the director n ˆ n0 ‡ dn? and the layer displacement u in an SmA liquid crystal.

Landau±Ginzburg free±energy density of superconductors [44]±[47]: gsc ˆ gn ‡ 12 fajcj 2 ‡ 12 bjcj 4 ‡ …2me †ÿ1 j…ÿip` ÿ 2eA†cj 2 ‡ …2m†ÿ1 …`  A† 2 ÿ H0 …`  A† ‡ 12 mH02 g;

…10:11†

where c corresponds to the wave function describing the Bose condensate of Cooper pairs and A is the magnetic vector potential. Due to this similarity, the mass density variation in smectic phases can be considered as an analog to the quantum mechanical wave function, and the magnetic ¯ux density `  A caused by a vector potential A is an analog to a bend or twist deformation of the director ®eld due to mechanical deformation or due to the chirality of the molecules (Table 10.1). Moreover, one can say that the smectic phase expels twist deformations like the superconducting Meissner phase expels an external magnetic ®eld (the Meissner±Ochsenfeld e¨ect [48], Figure 10.5(a), (b)). The most important consequence of this beautiful analogy was the prediction by de Gennes [14] that two types of liquid crystal materials should exist, and that a new state should appear in one of them. Superconductors of type II show a mixed state between the Meissner phase and the normal conducting state (Figure 10.6). Metal oxides which show the superconducting behavior of type II are of very high interest because of their high critical temperatures [49] which could be extended up to 130 K [50]. The mixed state was already discovered by Shubnikov in 1937 [18]. Abrikosov

10. Twist Grain Boundary Phases

305

Table 10.1. Analogy between the characteristic quantities in a smectic liquid crystal (left) and a superconductor (right), according to de Gennes [14] and Renn and Lubensky [1].

[19] explained this state as a regular lattice of normal conducting defects occurring in the superconductor (Figure 10.5(c)), a structure which can be very convincingly demonstrated by a decoration method [51]. In the core of the defects, the order parameter drops to zero and the magnetic ®eld can penetrate. The defects are stable if the surface energy, which is needed to ``melt'' the Bose condensate at the superconducting/normal-conducting interface, is compensated by the gain in magnetic energy. This is the case if the coherence length x of the wave function is su½ciently small in comparison with the London penetration length l, pin  particular, if the Ginzburg±Landau parameter k ˆ l=x is larger than 1= 2. Superconductors pof type I show no Shubnikov phase and are characterized by k < 1= 2, whereas superconductors p of type II show the Shubnikov phase and are characterized by k > 1= 2. In analogy, a smectic coherence length x can be de®ned for liquid crystals (Table 10.1). The extent to which an external deformation can penetrate into the smectic phase can be described by pa twist penetration length l2 . Consequently, liquid crystals with l2 < x= 2 (type I) should show a direct transition from the smectic phase (the analog of the Meissner phase) to the nematic or cholesteric phase (the analog of the normal conducting p state), whereas liquid crystals with l2 > x= 2 (type II) should show an inter-

306

H.-S. Kitzerow

Figure 10.5. (a), (b) Meissner e¨ect in superconductors: An external magnetic ®eld induces surface currents which result in a compensation of the external ®eld. (a) Current and magnetic ¯ux lines due to the induced magnetic dipole moment. (b) Flux lines of the resulting magnetic ®eld. (c) Abrikosov ¯ux lattice in type II superconductors: The external ®eld is not completely expelled, but penetrates in the superconductor along parallel tubes which form a regular lattice.

mediate state exhibiting a regular arrangement of defects in the smectic phase. In conclusion, TGB phases are the liquid crystal analog of the superconducting Shubnikov phase. Each defect in the Abrikosov lattice corresponds to a vortex † of the magnetic vector potential bearing one magnetic ¯ux quantum, A
ds ˆ mh=2e ˆ m
2:068
10ÿ15 V s …m ˆ 0; 1; 2; . . .†. In analogy, the modulus b of the Burger's vector of a dislocation in the TGB phase† must correspond to the layer spacing d or an integer multiple of d, so that n
ds ˆ md (Figure 10.7). In spite of the formal similarity between the phase diagrams of superconductors and liquid crystals, the range of parameters is much larger in the latter systems. Due to the anisotropy (which was neglected in the above summary), di¨erent penetration depths (lk and l? ) and di¨erent coherence lengths (xk and x? ) occur [12]. Moreover, the values of the parameter C? are not restricted to positive values (like the mass me ) but can also become negative. The values of C? < 0 correspond to the appearance of a tilt angle y between the director and the layer normal, as in the SmC phase. Theoretical consideration of the phase diagram in the …a; C? †-plane has shown [5] that the TGB phase is preferentially expected close to a N*±SmA±SmC* point because l2 diverges along the SmA±SmC* boundary and at the C? ˆ 0 line in the N* phase. For C? < 0, another phase, the TGBC phase, becomes stable which consists of SmC-like slabs …y 0 0† [22]. If both K11 and K22 are

10. Twist Grain Boundary Phases

307

Figure 10.6. Phase diagrams of (a), (b) superconductors and (c), (d) liquid crystals, according to de Gennes [14]. The analog of the external magnetic ®eld is the chirality ®eld h :ˆ K22 q0 .

larger than K33 , even a TGBC  phase with a helical director ®eld within each slab is expected to occur in the temperature range between SmC* and TGBC [26]. A further extension of the Renn±Lubensky theory predicts the appearance of a chiral line liquid NL in which the dislocations are disordered [30]. The latter state is the analog to the superconducting ¯ux line liquid [52]±[54]. The theory of TGB phases stimulated considerations on the elasticity and the interaction between dislocations, as well [55]. Not only the phenomenological theory, but also computer simulations using a Monte Carlo method [56] and using molecular-dynamics [57] con®rm the occurrence of TGB phases.

308

H.-S. Kitzerow

Figure 10.7. Edge dislocations in a smectic liquid crystal. The value of the path integral of n
ds along a closed loop is always an integer which corresponds to the number of inserted layers. According to de Gennes [14], this is the liquid crystal analog of the magnetic ¯ux quantization in superconductors.

10.3

Chemical Compounds Exhibiting TGB Phases

The ®rst series of compounds in which the appearance of the TGBA phase was proved was the homologuous series of …R† or …S†-1-methylheptyl-4 0 (4 00 -n-alkoxyphenyl-propioloyloxy) biphenyl-4-carboxylates (abbreviated 1M7nOPPBC or nP1M7), synthesized by Goodby et al. [2], [3]. The homologues with short chain length, n U 12, show an SmA±I transition, whereas the homologues with n V 16 show an SmC*±I transition. The TGBA phase occurs for the compounds with N ˆ 13, 14, or 15 just below the clearing point. In an …n; T†-phase diagram of this homologuous series, the TGBA phase appears close to the virtual SmA±SmC*±I meeting point. The transition enthalpy at the clearing point was found to be rather low in the homologues showing a TGB phase (A1 kJ/mol) and a broad nonsingular DSC peak indicates the occurrence of a further new phase in the isotropic region, which was called ``isotropic ¯ux phase'' or ``fog phase'' (because of its similarity to the blue fog BPIII) [58]. This additional phase is possibly identical with the twisted line liquid NL [30]. The racemic mixtures of these chiral

10. Twist Grain Boundary Phases

309

compounds show an SmA phase instead of TGBA , an SmC phase instead of SmC*, and no TGB nor NL phase [7]. The predicted phase sequence SmA±TGBA ±N* was ®rst observed in a mixture of cholesteryl nonanoate and nonyloxybenzoic acid [59], later in a chiral derivative of 4-biphenyl-benzoate [7] and many other pure compounds. According to the theoretical predictions, the occurrence of TGB phases requires a high optical purity and the abilities of the molecules to form a small cholesteric pitch as well as a smectic layer structure. The phase diagrams of chiral±racemic mixtures (Figure 10.8) con®rm that the TGB

Figure 10.8. Chiral±racemic phase diagram showing the dependence of the transition temperatures on the concentration ratio of two enantiomers (from [61]).

310

H.-S. Kitzerow

phases appear only for su½ciently high enantiomeric excess [7], [60], [61]. Attempts to explore the relation between the additional requirements and the molecular structure show that the balance of di¨erent properties is rather di½cult. In a ``pictorial approach'' to the property±structure correlations, Goodby [62] points out that the ability to form a smectic layer structure is supported by an extended molecular core of three or more benzene rings, while the ability to form a twisted structure with small pitch is supported by a zigzag structure of the molecule, combined with an internal ``molecular twist.'' Systematic variations of the phenylpropiolate derivatives [58] show that variations of the alkyl groups at the chiral center either make the chiral group more symmetric (e.g., when CH3 is replaced by C2 H5 ) or make the system more ¯exible (if the chain is too long) and is not suitable to enhance the TGB stability. However, a chloro-substitution in the chiral chain leads to compounds with the phase sequence SmA±TGBA ±N*±I. Replacing the carboxyl group which links the chiral chain to the aromatic core by a carbonyl group increases the polarity and stabilizes the SmA phase at the expense of the TGBA phase. Substituting the ether link of the n-alkyl chain in the nP1M7 series by a thioether link results in compounds which show no TGB phases. This can be attributed to the higher ¯exibility of the linking group [62]. It may also be speculated that the carboxyl group in the core which is present in many TGB compounds plays an important role by bending the overall structure of the molecule. Introducing ring substituents like a lateral ¯uor atom leads to the occurrence of TGBC phases, e.g., in the derivatives of 3-¯uoro-phenyl-propiolates [58]. The TGBC phase was observed for the ®rst time in the homologous series of 3-¯uoro-4-[…R† or …S†-1-methylheptyloxy]-4 0 -(4 00 -alkoxy-2 00 ,3 00 di¯uorobenzoyloxy)tolanes (nF2 BTFO1 M7 , Table 10.2) [23]. Careful calorimetric studies [63] have shown that even two di¨erent TGBC phases occur for some homologs of this series. Variation of the number of the lateral F atoms and their positions leads to the series of 3-¯uoro-4-[…R† or …S†-1methylheptyloxy]-4 0 -(4 00 -alkoxy-3 00 -¯uorobenzoyloxy)tolanes (nFBTFO1 M7 ) [64] and 2,3-di¯uoro-4-[…R† or …S†-1-methylheptyloxy]-4 0 -(4 00 -alkoxybenzoyloxy)tolanes (nBTF2 O1 M7 ) [65] which show TGBC and TGBA phases, as well. Nguyen et al. have extensively studied the dependence of the mesogenic behavior on the linking group between the aromatic core and the chiral side chain. The TGBA phase was found in many chiral derivatives of 4 0 -(4 00 alkoxybenzoyloxy)tolan-4-carboxylates [66]±[68] (e.g., compound 3d in Table 10.2). In these series, the chiral side chain is linked to the aromatic core by an ester group. No TGB phases occur if the direction of the ester group of the compounds (3d ) is reversed or if it is simply replaced by an ether link. In the case of an ether linking group, the lateral ¯uorine atom in the ortho position is a precondition for the formation of TGB phases [23]. It was concluded that a transverse dipole moment at the linking group is needed to stabilize the smectic A phase with respect to the cholesteric phase, and an

10. Twist Grain Boundary Phases Table 10.2. Some chemical compounds showing twist grain boundary (TGB) phases.

311

312

H.-S. Kitzerow Table 10.2 (continued )

10. Twist Grain Boundary Phases Table 10.2 (continued )

313

314

H.-S. Kitzerow

additional electron attracting e¨ect of the chiral group favors the formation of TGB phases [23], [64]. In addition, the steric hindrance due to lateral ¯uorine atoms is known to favor the formation of tilted phases in general [69], and thus supports the appearance of TGBC phases [36], [61], [70]. Although even small variations of the molecular structure can support or suppress the appearance of TGB phases, these mesophases have been observed in numerous liquid crystals (Table 10.2) [58]±[107]. The aromatic core of TGB substances contains not necessarily three benzene rings, as in the ®rst examples, but may also consist of two [71] or four benzene rings [41], [72], or contain a naphthyl unit [73]±[75]. The chiral chain may show two chiral centers [74]±[77]. The stability of a TGB phase can considerably di¨er in diastereomers [75]. It is interesting to note that the chirality of the molecules is not necessarily due to a chiral side chain, but may be due to saturated chiral rings forming the mesogenic core [78]±[80]. TGB phases also appear in mixtures containing chiral twin molecules where two mesogenic cores are linked by a chiral spacer [81]. In addition, they have been observed in dimesogens which consist of molecules with two mesogenic cores [82], [83], and trimesogens, i.e., molecules with three mesogenic cores [84]. Seshadri and Haupt [85] reported on the ®rst metallomesogen showing a TGB phase. There is also considerable interest in structures with an unsaturated alkyl chain [72], [73], [86] which are in principle suitable as polymer precursors. TGB phases also occur in side-chain cooligomers with a siloxane backbone [87]. However, the TGB phase found in a monomer with an unsaturated chain does not always appear in the respective side-chain polysiloxane [86]. Kozlovsky et al. [88], [89] have synthesized a large number of chiral side-chain polymers with a poly(methacrylate) or poly(siloxane) backbone. Their compounds show a TGB-like ``amorphous chiral smectic phase'' which is characterized by a smectic layer structure but vanishing birefringence [90], [91]. Copolymers of this type containing an azo group can be applied for optical storage [92], [93]. Of course, TGB phases are not only found in pure compounds but also in a large variety of mixtures [40], [59], [94]±[96], including systems composed of a nonchiral liquid crystal and a nonmesogenic chiral dopant [97]±[99]. Typically, TGB phases occur only in a narrow temperature interval of a few degrees Centigrade. However, a temperature range larger than 20  C was observed, as well [77].

10.4

Experiments on the TGBA Phase

An easy tool to identify liquid crystalline phases is the observation of characteristic textures in the polarizing microscope. A very characteristic pattern for TGB phases is the ®lament texture (Figure 10.9(a)) which occurs for anchoring of the director perpendicular to the surface of the sample (homeo-

10. Twist Grain Boundary Phases

315

tropic anchoring). If an SmA phase occurs in the temperature range below the TGB phase, the sample appears black between crossed polarizers in the SmA temperature range. On heating, small bright ®laments start growing at the SmA±TGB transition. To distinguish the TGB phase from a cholesteric phase requires some experience, but the curved shape of the TGB ®laments looks di¨erent from the cholesteric ®laments which can occur under similar conditions. Moreover, cholesteric ®laments appear typically only in a very narrow temperature range close to the SmA±N* transition. For parallel anchoring of the director, the TGB phase resembles the Grandjean texture of cholesteric phases. If a compound shows an N*±TGBA transition, its only indication is a slight widening and smearing out of the ``oily streaks'' which are characteristic for the N* phase [38]. In a side-chain oligomer, two different types of ®laments have been observed [87]. In wedge cells with parallel anchoring, the TGBA phase shows Grandjean± Cano (GC) lines [108], [109] like the cholesteric phase (Chapter 2). However, additional defect lines appear between neighboring w-disclinations (Figure 10.9(b)). Isaert et al. [110] attributed these additional GC lines to slab dislocations. In the TGBA phase, the slab dislocations were observed only for very small sample thickness, so far. This indicates that the line energies associated with the occurrence of the dislocations cannot be neglected. Thus, a quantitative analysis similar to the considerations in Chapter 2 was not possible because it is not clear whether the modulus of the Burger's vector corresponding to the observed dislocations is equal to the slab thickness lb or a multiple of lb [110]. However, the situation is di¨erent in the TGBC phase (Section 10.5). TGB samples with homeotropic anchoring or TGB droplets with a free surface can show a ®ngerprint texture which is typical for helical structures [111]. The subsequent appearance of concentric ®ngerprint lines in the SmC* phase, radial ®ngerprint lines in the TGBA phase, and radial ®ngerprint lines in the N* phase were observed in the appropriate liquid crystal droplets on increasing temperature [111]. This observation con®rms that the orientation of the pitch axis h with respect to the director n is the same in the TGBA phase as in the cholesteric phase, h ? n. An overview of TGB textures was published very recently [169]. The combination of cholesteric and smectic properties, which is apparent in TGB phases, can be clearly seen by the microscopic observation if the con®ning surfaces of the upper and lower glass plate are treated in order to give di¨erent alignments (Figure 10.9(c)) [112]. A cholesteric-like Grandjean texture appears if both substrates provide parallel alignment, whereas a dark area reminiscent of a SmA or SmC phase appears at the same temperature at locations where both substrates provide homeotropic alignment. However, ®laments appearing in the latter structure distinguish the TGB phase from other phases. Some interesting e¨ects have been observed under special conditions. By

316

H.-S. Kitzerow

(a)

(b) Figure 10.9. Some textures of TGB phases in a polarizing microscope. (a) Filament texture occurring at the SmA±TGBA transition in samples with homeotropic anchoring (courtesy of Dr. T. Seshadri [85]). (b) TGBA phase in a cell showing four regions with di¨erent surface treatment of the two substrates. From the top to the bottom: (1) homeotropic anchoring/parallel anchoring; (2) homeotropic/homeotropic; (3) parallel/homeotropic; (4) parallel/parallel (from [112]). (c) Grandjean± Cano lines in a wedge cell with parallel anchoring (from [110]). (d) Square pattern occurring in the TGBC  or undulated UTGBC phases (from [28]).

10. Twist Grain Boundary Phases

317

(c)

(d) Figure 10.9 (continued )

applying an electric ®eld, the ®laments can be aligned in the direction parallel to the ®eld [112]. Pattern formation at the TGB±isotropic interface can be induced by moving the sample along a temperature gradient [113]. Eventually, a TGB texture similar to the appearance of columnar phases can be observed [77]. This texture was attributed to the occurrence of microcolumns existing in the grain boundary regions [114]. These microcolumns are expected only if the slab thickness lb can be varied much easier than the dislocation distance ld [114]. The director ®eld of the TGBA phase is very similar to the director ®eld of a cholesteric phase (Chapter 2). Thus, a TGBA phase with parallel surface alignment of the director shows selective re¯ection of circularly polarized

318

H.-S. Kitzerow

(a) Figure 10.10. (a) Wavelength l of selective re¯ection versus temperature for the TGBA phase in the compound 14P1M7 (from [6]). Inset: Spectra for left- and righthanded circularly polarized light. (b) Wavelength l versus T for the TGBC and TGBA phases in the compound FH/FH/HH±13BTMHC (from [68]).

light if the pitch is of the same order of magnitude as the wavelength of the light. As in the N* phase, the Bragg wavelength is given by l0 ˆ pn cos y;

…10:12†

where y is the angle between the pitch axis and the direction of light propagation. Due to the vicinity of a smectic phase occurring at lower temperatures, the pitch p of TGB phases usually increases with decreasing temperature [6], [64], [96] (Figure 10.10). However, in several systems showing a reentrant cholesteric phase [115]±[118], a TGB phase occurs in the temperature range below a smectic phase. In this case, the temperaturedependence of the pitch can be reversed (i.e., the pitch p increases with increasing temperature).

10. Twist Grain Boundary Phases

319

(b) Figure 10.10 (continued )

In spite of the similarities, there is a fundamental di¨erence between the director ®elds of the N* and TGBA phases: The director ®eld of the cholesteric phase shows a continuous twist, whereas the TGB structure behaves optically like a stack of discrete birefringent blocks of ®nite thickness which have a uniform orientation of the optical axis [119]±[121]. The occurrence of additional selective re¯ection bands and unusual states of polarization are predicted for such a structure [119], [120]. In a commensurate phase, the respective wavelengths for back scattering …y ˆ 0† are approximately given by [120]: ll; G ˆ pn=…12 lN G 1†;

…10:13†

where N is the number of blocks per pitch, and l is an integer. The bandwith of the bands with higher order …l > 0† is expected to be rather small (a few nm). According to the sign appearing in the denominator of (10.13), two series of bands can occur for which the sign of the ellipticity is expected to be di¨erent, too. Since the number of slabs per pitch N is rather large for the commensurate TGBA phase, the wavelengths of higher order are below the visible wavelength range. Nevertheless, the observation of a higher-order selective re¯ection band was reported [120], although its spectral width was much larger than calculated. In agreement with the expectation that the same TGB structure can show a re¯ection of both right- and left-handed polarized light, Chilaya et al. [122] have observed an unusually low intensity of transmitted light (A20%) in the wavelength range of a selective re¯ection band, due to the presumed appearance of a TGB phase. In addition to the

320

H.-S. Kitzerow

unusual selective re¯ection, peculiar di¨raction patterns are expected for light incident perpendicular to the twist axis [119]. The most signi®cant information about the structure of TGB phases is obtained from high-resolution X-ray di¨raction studies of well-aligned samples [1], [6], [8], [24]. The structure factor of the TGBA phase has a cylindrical shape [1], [6]. If the helical axis h is oriented along the x-axis, the di¨raction pattern in the … y; z†-plane is a ring (see the inset of Figure 10.11(a)). The intensity pro®le of this ring along the radial direction is a dfunction, i.e., its ®nite widths in the experiment (Figure 10.11(a)) is due to the limited resolution of the experimental setup. This indicates a long-range smectic order along the director. However, positional ¯uctuations can in principle destroy the long-ranged translational order, leading to algebraic (rather than d-function) singularities, if a is irrational [123], [124]. The smectic order within each slab is described by a wave vector Qk with jQk j ˆ 2p=d. The scattering intensity of the ring along the concentric direction in the … y; z†-plane (w-scan) can answer the question whether a TGB phase is commensurate or incommensurate, i.e., whether the ratio lb = p of the smectic slab thickness and the pitch is a rational number or not. For an incommensurate phase [6], the scattering intensity of the ring is independent of the azimuthal angle w. However, for a commensurate phase [8], the ring consists of a ®nite number of distinct spots (Figure 10.11(c)). The cylinder in the reciprocal space which describes the structure factor (inset of Figure 10.11(a)) is broadened along the pitch axis due to the ®nite thickness lb of the smectic slabs. Thus, the scattering intensity I …Q? † in a plane parallel to the helix axis (o-scan) is described by a Gaussian of width xÿ1 A 2p…p
d†ÿ1=2 (Figure 10.11(b)). From the width of this Gaussian function, the thickness lb of the smectic slabs can be estimated [6]. The di¨raction pattern of an incommensurate TGBA phase di¨ers from a cholesteric phase with short-range smectic ordering by the radial width of the di¨raction ring appearing in the … y; z†-plane. As mentioned above, the scattering intensity is a d-function for the TGB phase. However, a Lorentzian line shape with width xÿ1 2 is expected for the cholesteric phase [1]. The twist lattice correlation length x2 diverges at the N*±TGBA transition. The X-ray data which are presently available seem to indicate that both incommensurate and commensurate TGBA phases can occur. An incommensurate TGBA phase with a pitch of p A 0:63±0.38 mm and a layer spacing of d A 4:3 nm was found in 14P1M7 [6]. Assuming lb A ld , the slab thickness was estimated to be lb A …d
p=2p† 1=2 ˆ 18:5 nm, which corresponds to Dw0 ˆ 360 lb = p A 13 [6]. However, a commensurate TGBA phase was found in 10BTF2 O1 M7 [8]. This compound shows a di¨raction ring consisting of 40 to 60 distinct spots (Figure 10.11(c)). The number of spots was found to decrease with increasing temperature. For the case of 46 spots, the reported parameters are d ˆ 3:8 nm, p ˆ 0:96 mm, lb ˆ p=46 ˆ 20:6 nm, Dw0 ˆ 360 =46 A 7:8 , and ld A d=Dw0 ˆ 27:8 nm [8]. It should be

10. Twist Grain Boundary Phases

321

Figure 10.11. X-ray di¨raction of the TGBA phase: (a), (b) Incommensurate TGBA phase: intensity distribution I…Q† in a plane (a) perpendicular and (b) parallel to the pitch axis (from [6]). (c) X-ray di¨raction pattern of a commensurate TGBA phase in the plane perpendicular to the pitch axis (from [8]).

322

H.-S. Kitzerow

(c) Figure 10.11 (continued )

stressed that lb and ld can be measured independently in a commensurate phase. In agreement with the theoretical expectation [1], the ratio of these two characteristic lengths is close to 1. A very convincing demonstration of the existence of smectic slabs in the TGBA phase was given by means of electron microscopy of freeze fractures [125]. Zasadzinski et al. have shown that the surface of freeze fractured samples consists of broad band-like regions which are striated with parallel lines (Figure 10.12(a)). The distance and direction of the parallel lines are constant within each of these regions but vary from region to region. The parallel lines can be interpreted as intersections of the smectic layers with the surface. The band-like regions correspond to the smectic slabs (Figure 10.12(b)). The width Lb of the uniformly striated regions is given by Lb ˆ lb =cos y

…10:14†

if y is the angle between the pitch axis h and the surface (Figure 10.12(c)). The periodicity of the parallel grooves at the surface may be described by a wave vector Qi with jQi j ˆ 2p=ai . Then, the wave vector Qi of the ith slab is given by the projection of the respective smectic wave vector Qi0 on the surface Qi ˆ Qi0 ÿ s0
…Qi0
s0 † ˆ 2p=d …ÿsin y cos wi ; sin wi ; 0†;

…10:15†

10. Twist Grain Boundary Phases

323

(a)

(b) Figure 10.12. Electron microscopy of freeze fractures. (a) Picture of the TGBA surface of 14P1M7 in the electron microscope. (b) Schematic drawing of the smectic blocks and the sample surface. (c) Orientation of the surface normal s with respect to the TGB structure (bottom) and the resulting pattern of the intersection lines between the surface and the smectic layers (top). (d) Experimental di¨raction spots obtained for the scattering of an HeNe laser beam on the surface pattern obtained by electron microscopy. (Parts (a), (b), and (d) of this ®gure are reprinted from [125].)

324

H.-S. Kitzerow

Q 0b /Q0

(c)

Q 0a /Q0 (d) Figure 10.12 (continued )

where s0 is a unit vector normal to the surface and wi is the azimuthal angle of Qi0 in the …y 0 ; z 0 †-plane normal to the pitch axis. Consequently, the respective distance ai between the parallel stripes is related to the smectic layer spacing d by ai ˆ d…1 ÿ cos 2 y cos 2 wi †ÿ1=2 :

…10:16†

10. Twist Grain Boundary Phases

325

Quantitative results were obtained by illuminating a picture obtained from electron microscopy (like Figure 10.12(a)) with an expanded laser beam and by studying the di¨raction pattern. The latter consists of Bragg spots, each corresponding to a reciprocal lattice vector Qi . The spots form an ellipse (Figure 10.12(d)), as expected from (10.15). For 14P1M7, Zasadzinski et al. [125] found a smectic layer spacing of d ˆ 4:1 G 0:3 nm, an angle between neighboring slabs of Dw0 A 17 , a distance between the screw dislocations of ld A d=Dw0 ˆ 14±15 nm, and a slab size of lb ˆ pDw0 =2p A 24±28 nm. These results are in good agreement with X-ray data (d A 4:3 nm, lb A ld A 18:5 nm, Dw0 ˆ 2plb = p A 13 [6]). In addition to electron microscopy, atomic force microscopy (AFM) was also used to study a TGBA phase. The latter study indicates a surface modulation which corresponds to the pitch of the director ®eld and was attributed to the elastic properties [106]. The behavior of the TGBA phase under high pressure [126]±[128] is quite di¨erent for di¨erent systems. In the pure compound 14P1M7, both the SmC*±TGBA transition and the transition from the TGBA phase to the isotropic phase were found to be shifted to lower temperature with increasing pressure [126]. The same e¨ect was found [128] for the SmC*±TGBA transition in the compound 12FBTFMO1M7. However, the slope of the transition line in the …p; T†-phase diagram changes sign above 200 and 600 bar, respectively. Above these critical values of the pressure, the temperature interval of the appearance of the TGBA phase becomes narrower. Finally, the TGBA phase disappears at very high pressures where the SmC* or SmA phase is induced. It is interesting to note that the slope d p=dT is negative for small pressures. According to the Clausius±Clapeyron equation d p=dT ˆ DH=TDV , this seems to indicate that the TGBA phase has a smaller volume than the SmC* phase which occurs at lower temperatures. This anomaly is similar to that of the ice±liquid transition in water. However, the transition enthalpies for the SmC* transition are very small. Thus, it may be more appropriate to explain the behavior by the properties of a second-order transition. In the latter case, the slope d p=dT is given by the changes of the compressibility w and the thermal expansion a: d p=dT ˆ Da=Dw ˆ Dcp =…TV Da†. In contrast to the detailed calorimetric studies mentioned above, there is still a lack of dilatometric studies in order to understand the …p; T†-phase diagrams precisely. Another interesting behavior under elevated pressures was observed [127] in binary mixtures of 4-(2 0 -methylbutyl) phenyl 4 0 -n-octyl biphenyl-4carboxylate (CE8) and 4-n-dodecyloxy biphenyl-4 0 -(2 0 -methylbutyl) benzoate (C12) which show TGB phases close to a virtual N*±SmA±SmC* triple point. Krishna Prasad et al. observed, in a mixture which shows a direct SmA±N* transition, that the appearance of the TGBA phase between SmA and N* can be induced by pressure. From the topology close to the SmA± TGB±N* point, it was concluded that this point is a critical end point rather than a bicritical point as predicted by Renn and Lubensky [5]. In a mixture with di¨erent concentrations which shows a SmC*±TGBA transition, the

326

H.-S. Kitzerow

appearance of an SmA phase was observed under elevated pressure. The resulting SmC*±SmA±TGBA meeting point was considered as a bicritical point.

10.5

Experiments on TGBC Phases

Twist grain boundary-C (TGBC ) phases poccur if the Landau±Ginzburg parameter k is not only smaller than 1= 2, but even negative (a situation which cannot occur in superconductors because it would correspond to a negative mass). Soon after the theoretical prediction [22], Nguyen et al. [23] found the ®rst experimental evidence for a TGBC phase in the chiral series of 3-¯uoro-4-[(R)-1-methylheptyloxy]-4 0 -(4 00 -alkoxy-2 00 ,3 00 -di¯uorobenzoyloxy) tolanes (nF2 BTFO1 M7 ). TGBC phases consist of a helical arrangement of smectic slabs like the TGBA phase, but the director n is tilted with respect to the layer normal N :ˆ Q=jQj (as in the smectic-C phase). The tilt direction is an additional parameter, and thus di¨erent structures of this kind are possible, even if the director is uniformly oriented within each slab. In the following, we use the notation suggested by Luk'yanchuk [29]. In the TGBCp phase discussed by Renn and Lubensky [22], both the director n and the layer normal Q are perpendicular to the pitch axis h (Figure 10.13(a)). The director ®eld and the density wave vectors Q i are essentially the same as in the TGBA phase. However, it is well known [129] that SmC*-like ordering leads to a spontaneous polarization Ps z Q  n. If both n and Q are perpendicular to h, the spontaneous polarization of each slab is parallel to the helical axis h, which is marked by the index ``p.'' If the tilt direction is the same for all smectic slabs, their polarizations add to give a macroscopic spontaneous polarization. Thus, the TGBCp structure (Figure 10.13(a)) corresponds to a true ferroelectric phase (as opposed to the helielectric SmC* phase where a macroscopic spontaneous polarization appears only in the unwound state). A hypothetical structure is represented in Figure 10.13(b), where the smectic layer normal is perpendicular to the helical axis, and the director is in the same plane as h and Q. In this case, the spontaneous polarization is perpendicular to the helical axis, and its azimuthal angle changes from slab to slab by Dw. Consequently, the macroscopic polarization is zero and the phase is helielectric, i.e., the spontaneous polarization can appear due to unwinding of the helical structure. However, X-ray investigations [24], [25] indicate the appearance of a di¨erent structure, the TGBCt phase (Figure 10.13(c)). The director of the TGBCt phase is perpendicular to the helix axis h, but the smectic layers are tilted within the plane de®ned by n and h. Again, Ps is perpendicular to h, and shows a helical arrangement so that the structure exhibits helielectric properties. The TGBCt structure was ®rst proposed by Dozov et al. [130], [131] and called the Melted Grain Boundary (MGB) phase in order to point out that the smectic order parameter essentially vanishes at the grain boundaries because the distance between the screw

Figure 10.13. Possible structures of TGBC phases and respective arrangements of the reciprocal lattice corresponding to the smectic blocks. (a) TGBCp : n ? h, Q s ? h, and Ps kh. (b) Hypothetical TGBC structure with Q s ? h, n tilted in the …Q s ; h†-plane, and Ps ? h. (c) TGBCt : n ? h, Q s tilted in the …n; h†-plane, and Ps ? h. (d) TGB2q : two superimposed TGBct structures with opposite tilt direction. (e) TGBC  : smectic slabs with an internal twisted SmC* structure (reprinted from [26]).

328

H.-S. Kitzerow Figure 10.13 (continued )

dislocations is very small. Note that the TGOCt structure di¨ers considerably from the two structures shown in Figure 10.13(a) and (b), because the reciprocal lattice vectors Qi form two cones instead of a planar orientation. This feature is in agreement with X-ray data [24], [25]. In addition to the structures presented in Figure 10.13(a)±(c), one can imagine intermediate structures where the spontaneous polarization has components along both the pitch axis and perpendicular to it [38]. However, another phase which di¨ers distinctly from these structures is the TGB2q phase (Figure 10.13(d)) which was predicted by Luk'yanchuk [29]. The TGB2q slab is a superposition of two equivalent SmC* populations with di¨erent tilt directions (``left'' and ``right'') of the layer normal with respect to the pitch axis. Finally, a TGBC  phase (Figure 10.13(e)) was predicted by Renn [26]. Instead of a uniform alignment of the director, the TGBC  phase shows a SmC*-like helical structure within each slab. Consequently, there is a superposition of N*-like and SmC*-like helical structures with the two local pitch axes perpendicular to each other. Moreover, the orientation of the smectic pitch axis changes from slab to slab. Due to this complicated structure, an additional type of coarse grain boundary is expected to occur in the TGBC  phase. The size lH of the coarse grains or helislabs is predicted to be much larger than the size of the smectic slabs lb and related to the latter by lH =lb ˆ …p=d† 1=2 .

10.5.1

Experimental Evidence and Properties of the TGB Ct Phase

In the TGBC phase of the compound 12F2 BTFO1 M7 (Table 10.2), Navailles et al. [24] discovered a commensurate structure for the ®rst time. The X-ray di¨raction pattern of this compound in a plane perpendicular to the helix axis is a continuous but strongly modulated ring (Figure 10.14(a), (b)). The number of the spots was found to decrease with increasing temperature from

10. Twist Grain Boundary Phases

329

(a)

(b) Figure 10.14. X-ray di¨raction of the uniformly aligned TGBCt phase of 12F2 BTFO1 M7 (from [24 and 25]). (a) Di¨raction pattern in the plane perpendicular to the pitch axis for a uniformly aligned sample. (b) Angular variation of the intensity along the ring of scattering (w-scan). (c) Di¨raction intensity in a plane parallel to the pitch axis (``o-scan''). Triangles: cholesteric phase. Squares: TGBC phase.

330

H.-S. Kitzerow

(c) Figure 10.14 (continued )

20 to 18, and ®nally to 16 which indicates that the phase is of the quasicrystalline type. The intensity scan in a plane parallel to the pitch axis (oscan) shows that the layers are not parallel to the pitch axis, but tilted by a tilt angle of oL U 14 (Figure 10.14(c)) which decreases continuously with increasing temperature [25]. This is in agreement with a structure of the type TGBCt . The characteristic lengths for the TGBCt structure of 12F2 BTFO1 M7 are [24] smectic layer spacing: d ˆ 3:75 nm, distance between the screw dislocations: ld ˆ d=Dw0 ˆ 11.8±9.4 nm (decreasing with increasing temperature), width of the smectic slabs: lb ˆ 100±65.6 nm, and pitch [23]: p ˆ 1±10 mm. In contrast to the TGBA phase, the ratio lb =ld di¨ers from 1, considerably …7 U lb =ld U 8:5† [24]. In compounds showing both a TGBA and TGBC phase, the latter appears in the lower temperature range. Adiabatic scanning calorimetry reveals a ®rst-order TGBC -TGBA transition (Figure 10.15) [102]. The wavelength of selective re¯ection of the TGBC phase is usually larger than that of the TGBA phase and shows a larger temperature-dependence (Figure 10.10(b)) [23], [104]. Due to the large pitch and the relatively large block size in TGBC phases, the observation of higher-order selective re¯ection bands (according to (10.13)) is more likely in TGBC phases than in the TGBA phase [120]. However, quantitative results are still missing. The origin of the commensurability is still unknown. One experimental approach to this question is the investigation of the elastic behavior of

10. Twist Grain Boundary Phases

331

Figure 10.15. Heat capacity of the compound FH/FH/HH±18BTMHC versus temperature, obtained by high-resolution adiabatic scanning calorimetry (from [102]).

deformed TGB phases. Two extreme cases may occur if the pitch changes due to external in¯uences: (1) the slab thickness lb can change very easily while the twist angle Dw, between neighboring slabs, is ®xed, Dw ˆ Dw0 ; or (2) the slab thickness lb remains constant and the twist angle Dw changes very easily. Galerne predicts that the ®rst condition may cause the existence of microcolumns in the grain boundary regions [114] and thus to the appearance of textures similar to columnar phases [77], whereas the second situation can lead to the surface-induced commensurability of TGB phases [132]. If the block size lb is ®xed (case 2), a strong parallel anchoring of the director at the substrates and a very small sample thickness L leads necessarily to a commensurate structure, because the conditions L ˆ np=2 (with n being an

332

H.-S. Kitzerow

integer) and L ˆ mlb (with m being an integer) are enforced, and thus p=lb ˆ 2m=n. However, careful studies of the X-ray intensities [133] and variations of the sample thickness [8] seem to indicate that the commensurability is an intrinsic property of certain TGB phases which is not exclusively induced by surface e¨ects. At least, ``integer values of the values of the ratio nb =n appear with a higher probability than noninteger values'' (where nb is the number of smectic blocks and n is the number of helical pitches in a cell with planar boundary conditions) [133]. The question, to what extent the directions of the smectic slabs of the TGBC phase are coupled, is also of interest because the model of melted grain boundaries [130], [131] seems to suggest that the smectic slabs are free to rotate. The investigation of the defect lines occurring in a Cano wedge (Figure 10.16) can give additional hints on this question. According to Isaert et al. [110], the elastic free-energy density corresponding to the dilation or compression of the TGB structure, due to the boundary conditions in a wedge cell, may be written as DgTGB ˆ 12 fB1 … p=p0 ÿ 1† 2 ‡ B2 …lb =lb; 0 ÿ 1† 2 ÿ 2B12 …p= p0 ÿ 1†…lb =lb; 0 ÿ 1†g 2 =B22 †… p= p0 ÿ 1† 2 ˆ 12 f…B1 ÿ B12

‡ B2 ‰…lb =lb; 0 ÿ 1† ÿ B12 =B2 …p=p0 ÿ 1† 2 Šg;

…10:17†

where p is the pitch, lb the slab thickness, and p0 and lb; 0 are the respective values of the unstrained material. The ®rst term in (10.17) with B1 ˆ K22 …2p= p† 2 corresponds to elastic twist deformation of the director ®eld, as in a cholesteric phase. In TGB phases, however, the slab thickness lb and the twist angle between neighboring slabs Dw may deviate from their equilibrium angle, in addition to elastic deformation of the pitch. The coupling parameter B12 in (10.17) describes to what extent a deformation of the pitch p is due to a change of the slab thickness lb or due to a change of the twist angle Dw. (Note that these quantities are related by (10.2)). For B12 ˆ B2 , any relative deviation … p=p0 ÿ 1† would correspond to the same relative deviation …lb =lb; 0 ÿ 1† at constant Dw. This situation corresponds to case (1) of the previous paragraph. In the other extreme case, B12 ˆ 0, p could be varied without a¨ecting lb , and vice versa. In this case, the free energy would be independent of Dw (case (2) of the above consideration). For minimum free energy [(10.16)], the number Nsd of slab dislocations between two neighboring disclination lines is given by [110]: Nsd … j† ˆ Int‰…1 ÿ B12 =B2 †f12 p0 =lb; 0 ‡ …M0 ÿ M†= jg  f1 ÿ …1 ‡ B12 =B2 † 2 =4 j 2 gÿ1 Š A 12 …1 ÿ B12 =B2 † p0 =lb; 0 ;

…10:18†

where j is the number of half-pitches in the observed section of the sample

10. Twist Grain Boundary Phases

333

Figure 10.16. (a) Schematic presentation of the dislocations occurring between two disclinations in a wedge cell with parallel anchoring (reprinted from [110]).

(Figure 10.16), M0 is the ratio j p0 =…2lb; 0 †, and M is the closest integer to M0 . For 12F2 BTFO1 M7 , the experimental values Nsd ˆ 5±6 and p0 =lb; 0 ˆ 16±20, yield the values B12 =B2 ˆ 0.5±0.75. This result indicates that the smectic slabs are not completely free to rotate, but show a favored angle Dw to some extent, i.e., the real situation is somewhere in between the two extreme cases (1) and (2). However, more precise studies are necessary, because (10.17) does not actually apply to commensurate TGB phases so that the result gives only qualitative hints.

10.5.2

The TGBC  Phase and the Undulated Twist Grain Boundary Phase

Experimental evidence for the occurrence of the TGBC  phase (Figure 10.13(e)) has been found in a mixture of 4,4 0 -di-heptyloxyazoxybenzene (HOAB) and 25 mol-% cholesteryl-benzoate [40]. A characteristic texture of this phase is the appearance of a square lattice (Figure 10.9(d)) [28]. A very similar pattern was observed in a di¨erent mixture and attributed to an Undulated Twist Grain Boundary phase, UTGBC  [27]. For parallel alignment of the director at the sample surface, this structure shows a square grid pattern in addition to the Grandjean±Cano (GC) lines. In a mixture of 4-(2methylbutyl phenyl)-4 0 -n-octyl biphenyl-4-carboxylate (CE8) and 2-cyano-4heptyl-phenyl-4 0 -pentyl-4-biphenyl [7(CN)5], the periodicity of the square

334

H.-S. Kitzerow

Figure 10.17. Model of the undulated UTGBC structure (from [27]).

lattice was found to be 2.5±4 mm, i.e., larger than the cholesteric pitch ( p ˆ 0.9±1.5 mm) but smaller than the distance between the GC lines in a wedge-shaped sample. The experimental observations indicate that the UTGBC  phase has indeed an SmC*-like helical structure within the slabs, as predicted by Renn and Lubensky. However, the appearance of the square grid was attributed to a periodic deformation (undulation) of the smectic slabs, which implies that the grain boundaries are not ¯at (as assumed in [26]) but undulating, too (Figure 10.17). Under the in¯uence of a 10 kHz electric ®eld the dark regions of the square grid become very thin and straight, which was attributed to the formation of solitons due to the ®eldinduced unwinding of the helical structure within each SmC* block [27].

10.5.3

Possible Observation of the TGB2q Phase

The TGB2q phase (Figure 10.13(d)) is expected to occur in the temperature range above the TGBCt phase [29]. The occurrence of this phase may be attributed to the observation of two TGBC phases, TGB1 and TGB2 , in the compound 8BTF2 O1 M7 (Table 10.2) [65]. X-ray studies have shown that the layers are tilted with respect to the helix axis in both of these phases. The low-temperature phase TGB1 shows a modulation of the X-ray intensity I …w† like the commensurate TGB Ct phase in other systems, whereas the hightemperature phase TGB2 seems to be incommensurate. Similarly, two TGBC phases, TGBCa and TGBCb , were found in the compounds 11F2 BTFO1 M7 and 12F2 BTFO1 M7 [39], [63], [134]. Again, the low-temperature phase TGBCa is the commensurate TGBCt phase which was investigated earlier [24], [25]. The high-temperature phase TGBCb appears only in a very narrow

10. Twist Grain Boundary Phases

335

temperature range of 0.05±0:1  C and was probably missed in the early works on TGBC phases. One may speculate that the phases TGB2 and TGBCb are identical to the predicted TGB2q phase. However, this is not yet con®rmed by experiment.

10.6

The Chiral Line Liquid

The chiral line liquid NL predicted by Kamien and Lubensky [30] is the analog of the ¯ux line liquid occurring in type II superconductors with strong ¯uctuations. Instead of forming a regular array, the defects are rather disordered. The position of the magnetic ¯ux lines in the superconducting ¯ux line liquid can be detected by means of magnetic force microscopy [135], [136]. However, the precise structure of the NL phase is still unknown. Calorimetric studies [31] show that an additional state occurs in liquid crystals between the TGBA phase and the usual cholesteric N* phase. The transition from the TGBA phase to this NL state is a ®rst-order transition with very small latent heat, e.g., A8:1 mJ/g ˆ 4:9 J/mol for 9FBTFO1 M7 [31] or <3 mJ/g for 10BTF2 O1 M7 [63]. However, the NL phase develops rather continuously into the N* phase on heating. The change from NL to N* is accompanied by a very broad peak of the heat capacity which extends over several K, but the comparison of measurements obtained with ac calorimetry and nonadiabatic scanning calorimetry for the same compound shows that no latent heat is associated with this transformation. The ®rst X-ray studies of 10BTF2 O1 M7 [8] have revealed a commensurate TGBA structure in the presumed TGBA temperature range and an incommensurate TGBA -like structure with quasi-long-range smectic order in the presumed NL temperature range. Due to this surprising latter result, more examples are needed to explore the characteristic features of the NL state. The compound 14P1M7 which shows a TGB±isotropic transition and no cholesteric phase, exhibits an additional highly disordered phase in a temperature range of about 4 K above the TGBA phase, which was detected by calorimetry [3], [4] and light-scattering experiments [137]. This additional state may also be a kind of twisted line liquid. However, the latent heat of the transition between the TGBA phase and this ``isotropic ¯ux phase'' is about 0.8±2.1 J/g [3], i.e., much larger than the latent heat of the TGBA ±NL transition in 9FBTFO1 M7 and 10BTF2 O1 M7 . Thus, the isotropic ¯ux phase must be much more disordered than the NL state appearing in the temperature range below a cholesteric phase.

10.7

Calorimetry and Phase Diagrams

High-resolution measurements of the heat capacity in di¨erent systems show that the topology of experimental phase diagrams (Figures 10.18, 10.19(b)) is similar to the theoretical prediction (Figure 10.19(a)). Variations of the

336

H.-S. Kitzerow

(a)

(b) Figure 10.18. (a) Phase diagram in the …n; T†-plane for the homologuous series of nFBTFO1 M7 . The number n corresponds to the alkyl chain length, noninteger values correspond to the mixture of two homologs (from [31]). (b) Schematic phase diagram of a superconductor showing a ¯ux line liquid (from [31]). (c) Magnetic force microscopy (MFM) image of vortices in a superconducting magnetic ¯ux line liquid (YBCO, image size 22 mm  22 mm). Top: Interaction between the vortices and the magnetic MFM tip (reprinted from [135]).

10. Twist Grain Boundary Phases

337

(c) Figure 10.18 (continued )

temperature and the alkyl chain length in a homologous series correspond to changes of the phenomenological parameters C? and a. As expected, the TGB phases occur in the vicinity of a virtual SmC*±SmA±N* multicritical point. However, in spite of the similarities, there are several di¨erences between the theoretical predictions and the experimental results. The current status can be summarized as follows [39]:

. The SmA±TGBA , TGBC ±TGBA , TGBA ±NL , and TGBC ±NL transitions

. . . .

were proved to be ®rst-order transitions whereas they were predicted to be second-order transitions. In analogy with superconductors, it can be assumed that ¯uctuations cause the two latter transitions to be of ®rst order. The TGBC stability range is typically much smaller than the TGBA range, whereas both ranges are theoretically expected to be similar. The range of existence of the TGB phases extends far from the virtual SmC*±SmA±N* point so that the critical end point, CEP, in Figure 10.19(a) was not observed, so far. Experiments seem to indicate that the SmC*±TGBC and TGBC ±TGBA transition lines meet tangentially, so that the junction point B3 in Figure 10.19 is not very well de®ned. Both experimental and theoretical work are necessary in order to clarify the structure of the TGBCb phase.

338

H.-S. Kitzerow

(a)

(b) Figure 10.19. (a) Theoretical phase diagram by Renn [26]. The cooordinates a and C? correspond to the parameters occurring in the Landau free energy (10.10). (b) Experimental phase diagram for the homologuous series of nF2 BTFO1 M7 (from [39]).

10. Twist Grain Boundary Phases

10.8

339

Electric Field E¨ects

A variety of e¨ects can occur in the TGB phases due to the in¯uence of an electric ®eld. The coupling between the director and the ®eld may be due to the dielectric anisotropy ea , or due to the dependence of the smectic tilt angle on the electric ®eld (electroclinic e¨ect), or due to the spontaneous polarization. In contrast to the typical behavior of smectic phases, a small electric ®eld cannot only result in a reorientation of the director, but also in a reorientation of the smectic layers [138]. Higher ®elds can cause a reorientation of the pitch axis, helical unwinding [139], [140], a shift of the wavelength of selective re¯ection [141], or ®eld-induced phase transitions [103], [141].

10.8.1

Electric Field E¨ects in the TGBA Phase

The behavior of the TGBA phase for small electric ®elds indicates an electroclinic e¨ect [103], [138], [142]±[144]. This e¨ect occurs in the chiral SmA phases due to the coupling between the spontaneous polarization Ps and the tilt angle y between the director and the layer normal [145]. Although y and Ps are zero in the SmA phase, an electric ®eld can induce a polarization thereby leading to a nonzero tilt angle y…E† ˆ ec E, where ec is the electroclinic coe½cient. In the uniformly aligned SmA phase, the free-energy density close to the transition temperature Tc of a second-order SmC*±SmA transition may be written as [129]: g ˆ 12 ay 2 ‡ 12 by 4 ÿ sEy;

…10:19†

where the ®eld-induced electric polarization is given by P ˆ sy. Minimizing the free energy given in (10.19) leads to ec ˆ s=‰a…T ÿ Tc †Š. In the TGBA phase, the situation is more complicated because each smectic slab exhibits a di¨erent angle wi between the electric ®eld and the layer normal. Besides this technical aspect, there is a fundamental di¨erence in the electroclinic behavior of the SmA phase and the TGBA phase. In the SmA phase, the electroclinic e¨ect causes a reorientation of the director while the layer normal is ®xed. However, a ®eld-induced reorientation of the layers was detected in the TGBA phase by X-ray investigation (Figure 10.20) [138]. Petit et al. [138] proposed a phenomenological description of this e¨ect which is shortly reviewed in the following. It is assumed that both the director n and the spontaneous polarization remain perpendicular to the pitch axis x if a ®eld is applied along the y-direction perpendicular to the pitch axis. Instead of a uniform tilt angle y, each slab may show a di¨erent ®eldinduced tilt angle oL; i . Thus, the last term in (10.19) is replaced by ÿPi
E ˆ ÿsoL; i E cos wi . A sixth-order term is added and the fourth-order term is neglected, taking into account a tricritical behavior …oL z jT ÿ Tc j 1=4 † which was experimentally observed. The elastic free energy of the director

340

H.-S. Kitzerow

Figure 10.20. Electroclinic e¨ect in TGB phases: electric ®eld-induced change of the X-ray di¨raction intensity for observation of the compound 11F2 BTFO1 M7 perpendicular to the pitch axis (``o-scan''): (a) TGBA phase; and (b) TGBC phase (from [138]).

10. Twist Grain Boundary Phases

341

®eld due to the twist angle …oL; i‡1 ÿ oL; i † between neighboring slabs is taken into account by an additional term proportional to …oL; i‡1 ÿ oL; i † 2 . This results in a free energy per unit area in the … y; z†-plane X 2 2 6 1 ‰AoL; gˆ i =2 ‡ CoL; i =6 ‡ 2 D…oL; i‡1 ÿ oL; i † ÿ soL; i E lb; i cos wi Š: slabs i

…10:20† Approximating the discontinuous structure by a continuum consisting of slabs with in®nitesimal thickness leads to … g A …2p†ÿ1 C ÿ1=2 D 3=2 fRF 2 =2 ‡ F 6 =6 ‡ 12 …dF=dw† 2 ÿ LF cos wg dw; …10:21† 1=4

where R ˆ A=D ˆ a…T ÿ Tc †=D is a reduced temperature, L ˆ EC slb =D is a reduced ®eld strength, and F ˆ C 1=4 Dÿ1=4 oL . The solutions of the corresponding Euler±Lagrange equation are in good qualitative agreement with the experimental data (Figure 10.21). In the temperature range of the TGBA phase, a particular solution is given by oL ˆ ‰slb cos w=fD ‡ a…T ÿ Tc †gŠE:

…10:22†

The induced tilt angle depends linearly on the applied ®eld, but the electroclinic coe½cient oL =E is modulated between 0 (for w ˆ Gp=2 , nkE) and a maximum value (for w ˆ 0 , n ? E), due to the helical structure of the TGB phase. Investigations using dielectric spectroscopy [143] show a second molecular relaxation process besides the soft mode. This additional mode was attributed to a switching process restricted to the vicinity of grain boundaries ( process due to grain boundaries, PGB). It is observed only for thick samples and vanishes if the helix is unwound. In 14P1M7, the relaxation frequency for the soft mode was found to be 10±100 kHz, the relaxation frequency for the PBG process A 1 kHz [143]. NMR relaxation studies of a TGBA phase give additional hints that the molecular dynamics in TGBA phases is clearly di¨erent from the conventional SmA phase of chiral molecules [146]. Optical investigations of 14P1M7 with parallel anchoring and application of the ®eld along the pitch axis indicate several e¨ects [141]. Using crossed polarizers, a modulation of the transmitted light intensity could be detected which depends quadratically on the ®eld strength. In addition, a shift of the wavelength of selective re¯ection is observed. For higher ®elds, the Grandjean texture becomes unstable and the ®lament texture of the TGBA phase is induced. Finally, the SmC* phase is induced which is stabilized by the electric ®eld due to its spontaneous polarization. Since the compound 14P1M7 shows a direct transition from the SmC* phase to the TGBA phase even without a ®eld, this ®eld-induced transition can be considered as a continuous shift of the transition temperature (Figure 10.22). Assuming that the electric polarization induced in the TGB phase can be neglected with respect

342

H.-S. Kitzerow

(a)

(b)

Figure 10.21. Electroclinic e¨ect in TGB phases: tilt angle of the smectic layers versus temperature for di¨erent electric ®eld strengths. (a) Experimental values of the compound 11F2 BTFO1 M7 , obtained from X-ray investigations. (b) Solutions of the minimization of the free energy given in (10.21), according to Petit et al. (from [138]).

to the spontaneous polarization Ps of the SmC* phase, the slope dT=dE of the transition line in the …E; T†-phase diagram can be described by a relation similar to the Clausius±Clapeyron equation dT=dE ˆ ÿ…PII ÿ PI †=…SII ÿ SI † ˆ TjPs j=DH;

…10:23†

where DH is the latent heat of the SmC*±TGBA transition. Equation (10.23) is in good agreement with the experimental observation, where dT=dE ˆ 4
10ÿ7 Km/V was found for Ps ˆ 10ÿ4 As/m 2 , T ˆ 363 K, and DH A 7


10. Twist Grain Boundary Phases

343

Figure 10.22. Electric ®eld-induced phase transition: phase diagram of 14P1M7 in the (electric ®eld, temperature)-plane (from [141]).

10 4 J/m 3 . Shao et al. [103] have found a similar ®eld-induced phase transition, namely a stabilization of the SmA phase with respect to the TGBA phase due to an electric ®eld. The threshold ®eld strength of this transition is decreasing with increasing temperature, and the transition is irreversible. Thus, it has been proposed for optical storage applications.

10.8.2

Electric Field E¨ects in the TGBC Phase

In the TGBC phase, each smectic slab exhibits a spontaneous polarization which is perpendicular to the pitch axis and shows a helical structure. Consequently, helielectric behavior can be expected. Indeed, a helical unwinding from the TGBC phase to a uniformly oriented SmC* phase was observed [103]. Recently [140], this e¨ect of helical unwinding was studied in more detail by investigating the X-ray intensity distribution along the ring of spots which are observed if the incident radiation propagates along the pitch axis of a uniformly aligned sample (Figure 10.23). In the ®eld-o¨ state, the w-scan shows a de®ned number of separated spots with nearly equal intensity (Figure 10.23(d)). A small variation of the amplitude could be explained by the superposition of two domains with 18 and 19 spots, respectively. Under the application of an electric ®eld perpendicular to the pitch axis, a clear sinusoidal modulation of the di¨raction intensity was observed (Figure 10.23(e)) with a maximum at w ˆ 0 (corresponding to slabs with Ps kE) and a minimum at w ˆ 180 (corresponding to slabs with antiparallel orientation of Ps with respect to E ). At the same time, the maxima became not clearly separated. At even higher ®elds, a ®nite number of spots was observed again with

344

H.-S. Kitzerow

Figure 10.23. Ferroelectric response of a TGBC phase (compound 12F2 BTFO1 M7 ): (a)±(c) Simulated and (d)±(f ) experimental X-ray di¨raction o-scans of the TGBC phase under the in¯uence of a dc electric ®eld perpendicular to the pitch axis. (a), (d) 0 V; (b), (e) 100 V; (c), (f ) 250 V (from [140]).

a nearly square-shaped envelope of the intensities (Figure 10.23(f )). This behavior indicates that smectic slabs with an orientation of Ps between ÿ90 and ‡90 with respect to E are stabilized, whereas slabs with an orientation of Ps between 90 and 270 are destabilized and even vanish at high ®elds. Finally, for E > 10 5 V/m only one peak at w ˆ 0 was detected, indicating the appearance of the SmC* phase [140]. The mechanism of these changes of the X-ray intensities can be attributed to di¨erent mechanisms. Either the slabs which are stabilized may grow while the slabs which are destabilized shrink, or the slabs are rotated around the pitch axis with lb remaining constant. Consequently, Barois et al. [140] consider the following free energy per unit surface area for a phenomenological description X f12 B2 …lb; i ÿ lb; 0 † 2 ‡ 12 B3 …wi‡1 ÿ wi ÿ Dw0 † 2 gˆ slabs i

ÿ B23 …lb; i ÿ lb; 0 †…wi‡1 ÿ wi ÿ Dw0 † ‡ D2 =4…lb; i ÿ lb; 0 † 4 ÿ Ps E lb; i cos wi g;

…10:24†

where lb; i and …wi‡1 ÿ wi † are the slab thickness and twist angle between

10. Twist Grain Boundary Phases

345

neighboring slabs under the in¯uence of the ®eld E, whereas lb; 0 and Dw0 are the respective values for E ˆ 0. Equation (10.2) is not valid for E 0 0. Again, two extreme cases can be considered: (1) B2 f B3 : the slab thickness lb; i varies easily, but the twist angle Dw is ®xed; or (2) B2 g B3 : the slab thickness is constant and the twist angle Dw is free to change. Barois et al. point out that the second case would require the creation or annihilation of dislocations and is less favorable than the ®rst case. The results of minimizing the free energy X f12 B2 …lb; i ÿ lb; 0 † 2 ‡ D2 =4…lb; i ÿ lb; 0 † 4 ÿ Ps E lb; i cos wi g; …10:25† gˆ slabs i

(assuming that Dw ˆ Dw0 ˆ constant) are in qualitative agreement with experimental data (Figure 10.23), except for a reduction of the contrast for medium ®eld strength (Figure 10.23(e)) which indicates that deviations of the angle Dw lead to a loss of commensurability in this medium range of voltages. Dielectric studies of the TGBA and TGBC phases [144] indicate two relaxation processes, a soft mode and a Goldstone mode, as in SmA and SmC* phases (Chapter 8). However, they appear with lower amplitude and at higher frequencies than in the SmA and SmC* phases, which is explained by additional elastic e¨ects due to the block structure with limited block thickness.

10.9

Smectic Blue Phases

It is interesting to note the similarities between blue phases (Chapter 7) and TGB phases. Both types of phases show a local structure [occurring in a volume of about (10 nm) 3 ] and a superstructure [with a periodicity of some hundred nm or a few mm] which are incompatible with respect to each other. The impossibility of matching these two structures by a continuous molecular arrangement with constant degree of order (frustration) leads to a regular lattice of defects. The type of defects (dislocations in TGB as opposed to disclinations in BP) depends on the order parameter which describes the transition between the two phases which occur in the temperature ranges below and above the ``mixed state.'' In the core of dislocations (TGB), the smectic order parameter c (10.7) vanishes, thereby enabling the penetration of twist. In disclinations (BP), the orientational order parameter vanishes, thereby enabling the occurrence of double twist in regions between the disclinations. In principle, both TGB and BP phases can be considered as heterogeneous systems consisting of the respective high-temperature and low-temperature phases [147]. Another similarity between TGB and BP is

346

H.-S. Kitzerow

the occurrence of a ``fog phase'' (NL and BPIII, respectively). The transition from the regular lattice of defects to the nonordered fog phase can be considered as an example of lattice melting [148]. The development from the fog phase to the high-temperature phase can happen continuously without a latent heat (NL ! N  [134], BPIII ! I [149]). In addition to these formal analogies, there is at least one blue phase modi®cation, BPS [150], [151], which shows smectic order [152]. This observation was con®rmed [153] for the compound FH/FH/HH±18BTMHC which exhibits a direct transition from the TGBA phase to a blue phase [154]. The X-ray di¨raction pattern in the BP temperature region is a continuous ring (sharp in the radial direction) exhibiting an intensity modulation with fourfold symmetry [155]. This result seems to indicate that TGB phases cannot only have a cholesteric superstructure, but also a BP-like cubic superstructure.

10.10

The Smectic-Q Phases

Due to the existence of TGBA and TGBC phases, it is obvious to ask whether antiferroelectric TGB phases exist. In 1983, structural studies [156] gave evidence for two new mesophases which were labeled smectic-O (SmO) and smectic-Q (SmQ). The appearance of these two phases is very sensitive to the chirality of the system. One of these phases, SmO, was shown to be identical to the antiferroelectric SmCA phase (Chapter 9) [157], [158]. Consequently, today the SmO phase is labeled as ``SmCA ,'' although the alternating tilt also occurs in nonchiral materials which are not antiferroelectric [159]. However, the second phase, SmQ, appears only in chiral liquid crystals. Recent X-ray investigations by Levelut et al. [33] indicate the occurrence of at least four di¨erent structures of SmQ phases which have to be considered as ``crystals of defects.'' In contrast to conventional TGB phases, they show a tridimensional lattice of dislocations (as opposed to parallel walls). The four identi®ed structures T*I, T*II, T*III, and H* are described by the space groups I422 (for T*I), I41 22 (T*II), P41 22 (T*III), and P62 22 (H*), respectively. The lattice constants of these structures are in the range of A6:5±13 nm. According to Levelut et al. [33], the T*II and H* structures can be considered as arrays of twist grain boundaries with twist angles of p=2 and p=3, respectively, in an antiferroelectric smectic liquid crystal. The T*III structure consists of two identical labyrinths with a local SmCA organization which are separated by a CLP mimimal surface. The body-centered tetragonal T*I structure consists of double twist clusters without smectic organization.

10.11

Bend Grain Boundary (BGB) Phases

Eleven years were necessary to recognize that the mixed state predicted by de Gennes [14] cannot only be induced by mechanical deformation of the director ®eld, but can even occur spontaneously due to the presence of chiral

10. Twist Grain Boundary Phases

347

Figure 10.24. Molecules with a bent molecular core are expected to show a high ¯exoelectric coe½cient. Thus, an electric ®eld-induced bend deformation may be suitable to generate a lattice of edge dislocations, thereby leading to a bend grain boundary phase.

molecules [1]. The experimental evidence for the occurrence of the mixed state in the form of TGB phases has initiated a revival of the search for regular defect structures in nonchiral materials due to mechanical forces [160]. In addition, we should think of a further challenge, namely to look for the mixed state in bent structures. The success of the helical ®eld h to induce twist of the director ®eld (10.4) makes it reasonable to think of an equivalent ®eld which induces bend. A good candidate is the electric ®eld E in connection with the e¨ect of curvature electricity (the ¯exoelectric e¨ect [161]). The change of the free-energy density due to the electric ®eld-induced ¯exoelectric polarization is given by gfl ˆ ÿEPfl ˆ ÿEfe1 ‰n…`
n†Š ‡ e3 ‰n  …`  n†Šg;

…10:26†

where e1 and e3 are the ¯exoelectric coe½cients of splay and bend, respectively. In particular, molecules with a bent core like the new ``bananashaped'' molecules [162]±[164] are expected to have a high bend ¯exoelectric coe½cient e3 (Figure 10.24). When the linear twist term ‡h…n
`  n† in (10.4) is replaced by the ¯exoelectric bend term ÿe3 E‰n  …`  n†Š, it becomes obvious that the electric ®eld may have a similar e¨ect on the bending of the director ®eld as the addition of chiral molecules on the twist. Qualitatively, the formal analogy between superconductors and liquid crystals (Table 10.1) may be applied to describe the case of a ®eld-induced bend deformation. In this case, the chirality ®eld h has to be replaced by the term ÿe3 E which corresponds to the electric ®eld, the twist elastic constant K22 has to be replaced by the bend elastic constant K33 , and the relevant smectic coe½cient is Ck instead of C? .

10.12

Conclusion

The extensive studies on TGB phases during the last decade have generated a very exciting new ®eld of liquid crystal research. Today, there are still many open questions: Do commensurate TGBA phases occur in many compounds? Which is the microscopic origin of the lock-in at certain angles w which leads to commensurate phases? Which is the precise structure of the NL state? Is the main di¨erence between TGBA and NL that one phase (TGBA ) is commensurate whereas the other (NL ) is incommensurate? Is

348

H.-S. Kitzerow

there an experimental method to make the positions of the defects in the NL phase visible (like detecting the magnetic ¯ux tubes in superconductors by magnetic force microscopy)? How does the chiral line liquid behave under the in¯uence of electric ®elds? Does the structure of the TGB2 or TGBCb phases correspond to the theoretically predicted TGB2q phase, and is the undulated TGB phase equivalent to TGBC ? How can the di¨erences between theoretical and experimental phase diagrams be explained [39]? Can the anomalous behavior of some compounds under pressure [126], [128] be explained by assuming second-order phase transitions although calorimetric studies indicate that the same phase transitions are of ®rst order in other compounds? Do TGB phases show anomalous attenuation of sound modes and vanishing sound speeds for certain directions of propagation although they are translationally ordered in all three dimensions [165]? Are TGB phases interesting for commercial applications, e.g., due to electro-optic e¨ects [141] or due to optical storage e¨ects [92], [93], [103]? In addition to the investigated materials, dislocation lattices may also occur in other systems, e.g.:

. . . .

columnar phases [166]; lyotropic lamellar phases [167]; polymers dissolved in chiral nematic liquid crystals [168]. Are other antiferroelectric TGB phases possible in addition to the smecticQ phases [33]? . Maybe, bend grain boundary (BGB) phases can be induced, indeed (Section 10.9). In conclusion, the research on twist grain boundary phases is still a challenge for both experimentalists and theorists, at least for another decade.

References [1] S.R. Renn and T.C. Lubensky, Phys. Rev. A 38, 2132±2147 (1988). [2] J.W. Goodby, M.A. Waugh, S.M. Stein, E. Chin, R. Pindak, and J.S. Patel, Nature 337, 449±452 (1989). [3] J.W. Goodby, M.A. Waugh, S.M. Stein, E. Chin, R. Pindak, and J.S. Patel, J. Am. Chem. Soc. 111, 8119±8125 (1989). [4] C.C. Huang, D.S. Lin, J.W. Goodby, M.A. Waugh, S.M. Stein, and E. Chin, Phys. Rev. A 40, 4153±4156 (1989). [5] T.C. Lubensky and S.R. Renn, Phys. Rev. A 41, 4392±4401 (1990). [6] G. Srajer, R. Pindak, M.A. Waugh, J.W. Goodby, and J.S. Patel, Phys. Rev. Lett. 64, 1545±1548 (1990). [7] A.J. Slaney and J.W. Goodby, Liq. Cryst. 9, 849±861 (1991). [8] L. Navailles, B. Pansu, L. Gorre-Talini, and H.T. Nguyen, Phys. Rev. Lett. 81, 4168 (1998). [9] T.C. Lubensky, T. Tokihiro, and S.R. Renn, Phys. Rev. Lett. 67, 89±92 (1991). [10] F.C. Frank, Disc. Faraday Soc. 25, 19 (1958).

10. Twist Grain Boundary Phases

349

[11] S. Chandrasekhar, Liquid Crystals, 2nd ed., Cambridge University Press, Cambridge, 1992. [12] P.G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed., Oxford University Press, Oxford, 1993. [13] L. Pohl and U. Finkenzeller, Physical Properties of Liquid Crystals, in: Liquid CrystalsÐApplications and Uses (edited by B. Bahadur), Vol. 1, Chap. 4, pp. 139±170, World Scienti®c, Singapore, 1990. [14] P.G. de Gennes, Solid State Commun. 10, 753 (1972). [15] L. Cheung, R.B. Meyer, and H. Gruler, Phys. Rev. Lett. 31, 349 (1973). [16] R.S. Pindak, C.-C. Huang, and J.T. Ho, Phys. Rev. Lett. 32, 43±46 (1974). [17] P.G. de Gennes, Mol. Cryst. Liq. Cryst. 21, 49 (1973). [18] Shubnikov, Khotkevich, Shepelev, and Rabinin, Zh. Eksp. Teor. Fiz. 7, 221± 237 (1937). [19] A.A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957); Sov. Phys. JETP 5, 1174 (1957). [20] W.U. MuÈller, Phasenumwandlungen und Strukturen cholesterischer ¯uÈssiger Kristalle, Thesis, Technical University Berlin (1974). [21] Pollmann, private communication. [22] S.R. Renn and T.C. Lubensky, Mol. Cryst. Liq. Cryst. 209, 349±355 (1991). [23] H.T. Nguyen, A. Bouchta, L. Navailles, P. Barois, N. Isaert, R.J. Twieg, A. Maarou®, and C. Destrade, J. Phys. France II 2, 1889±1906 (1992). [24] L. Navailles, P. Barois, and H.T. Nguyen, Phys. Rev. Lett. 71, 545±548 (1993). [25] L. Navailles, R. Pindak, P. Barois, and H.T. Nguyen, Phys. Rev. Lett. 74, 5224 (1995). [26] S.R. Renn, Phys. Rev. A 45, 953±973 (1992). [27] P.A. Pramod, R. Pratibha, and N.V. Madhusudana, Current Science 73, 761± 765 (1997). [28] W. Kuczynski and H. Stegemeyer, SPIE 3318, 90±93 (1997). [29] I. Luk'yanchuk, Phys. Rev. E 57, 574±581 (1998). [30] R.D. Kamien and T.C. Lubensky, J. Phys. France I 3, 2131±2138 (1993). [31] T. Chan, C.W. Garland, and H.T. Nguyen, Phys. Rev. E 52, 5000±5003 (1995). [32] B. Pansu, M.H. Li, and H.T. Nguyen, Eur. Phys. J. B 2, 143±146 (1998). [33] A.-M. Levelut, E. Hallouin, D. Bennemann, G. Heppke, and D. LoÈtzsch, J. Phys. France II 7, 981±1000 (1997). [34] J.W. Goodby, J. Mater. Chem. 1, 307 (1991). [35] J.W. Goodby, A.J. Slaney, C.J. Booth, I. Nishiyama, J.D. Vuijk, P. Styring, and K.J. Toyne, Mol. Cryst. Liq. Cryst. 243, 231±298 (1994). [36] A. Bouchta, H.T. Nguyen, L. Navailles, P. Barois, C. Destrade, F. Bougrioua, and N. Isaert, J. Mater. Chem. 5, 2079±2092 (1995). [37] T.C. Lubensky, Physica A 220, 99±112 (1995). [38] W. Kuczynski, Selforganization in Chiral Liquid Crystals, Scienti®c Publishers OWN, Poznan, 1997. [39] C.W. Garland, Liq. Cryst. 26, 669±677 (1999). [40] W. Kuczynski and H. Stegemeyer, Mol. Cryst. Liq. Cryst. 260, 377±386 (1995). [41] V. Vill, LiqCryst( Database, Version 3.1, LCI Publisher, Hamburg, 1998. [42] L. Landau, Phys. Z. Sowjet. 11, 545±555 (1937). [43] P.G. de Gennes, Mol. Cryst. Liq. Cryst. 12, 193 (1971). [44] V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1044 (1950).

350

H.-S. Kitzerow

[45] P.G. de Gennes, Superconductivity of Metals and Alloys, W.A. Benjamin, New York, 1966. [46] D.R. Tilley and J. Tilley, Super¯uidity and Superconductivity, Van Nostrand Reinhold, New York, 1974. [47] W. Buckel, SupraleitungÐGrundlagen und Anwendungen, 5th ed., VCH, Weinheim, 1993. [48] W. Meissner and R. Ochsenfeld, Naturwissenschaften 21, 787 (1933). [49] J.G. Bednorz, M. Takashige, and K.A. MuÈller, Europhys. Lett. 3, 379 (1987). [50] A. Schilling, M. Cantoni, J.D. Gao, and H.R. Ott, Nature 363, 56±58 (1993). [51] U. Essmann and H. TraÈuble, Phys. Lett. A 24, 526 (1967). [52] T. Ota, Y. Kimura, and R. Hayakawa, Ferroelectrics 178, 125±133 (1996). [53] D.R. Nelson, Phys. Rev. Lett. 60, 1973±1976 (1988). [54] P.L. Gammel, L.F. Schneemeyer, J.V. Waszczak, and D.J. Bishop, Phys. Rev. Lett. 61, 1666±1669 (1988). [55] Y. Hatwalne and T.C. Lubensky, Phys. Rev. E 52, 6240±6249 (1995). [56] R. Memmer, H.-G. Kuball, and A. SchoÈnhofer, Liq. Cryst. 15, 345±360 (1993). [57] M.P. Allen, M.A. Warren, and M.R. Wilson, Phys. Rev. E 57, 5585±5596 (1998). [58] J.W. Goodby, I. Nishiyama, A.J. Slaney, C.J. Booth, and K.J. Toyne, Liq. Cryst. 14, 37±66 (1993). [59] O.D. Lavrentovich, Y.A. Nastishin, V.I. Kulishov, Y.S. Narkevich, and A.S. Tolochko, Europhys. Lett. 13, 313±318 (1990). [60] C.J. Booth, D.A. Dunmur, J.W. Goodby, J.S. Kang, and K.J. Toyne, J. Mater. Chem. 4, 747±759 (1994). [61] C.J. Booth, J.W. Goodby, K.J. Toyne, D.A. Dunmur, and J.S. Kang, Mol. Cryst. Liq. Cryst. 260, 39±50 (1995). [62] J.W. Goodby, Mol. Cryst. Liq. Cryst. 292, 245±263 (1997). [63] L. Navailles, C.W. Garland, and H.T. Nguyen, J. Phys. France II 6, 1243± 1258 (1996). [64] A. Bouchta, H.T. Nguyen, M.F. Achard, F. Hardouin, and C. Destrade, Liq. Cryst. 12, 575±591 (1992). [65] L. Navailles, H.T. Nguyen, P. Barois, N. Isaert, and P. Delord, Liq. Cryst. 20, 653±664 (1996). [66] H.T. Nguyen, R.J. Twieg, M.F. Nabor, N. Isaert, and C. Destrade, Ferroelectrics 121, 187±204 (1991). [67] L. Navailles, H.T. Nguyen, P. Barois, C. Destrade, and N. Isaert, Liq. Cryst. 15, 479±495 (1993). [68] M.-H. Li, V. Laux, H.-T. Nguyen, G. Sigaud, P. Barois, and N. Isaert, Liq. Cryst. 23, 389±408 (1997). [69] K. Furukawa, K. Terashima, M. Ichihaski, S. Saitoh, K. Miyazawa, and T. Inukai, Ferroelectrics 85, 451 (1988). [70] H.T. Nguyen, A. Babeau, J.C. Rouillon, G. Sigaud, N. Isaert, and F. Bourgrioua, Ferroelectrics 179, 33±50 (1996). [71] H.T. Nguyen, C. Destrade, J.P. Parneix, P. Pochat, N. Isaert, and C. Girold, Ferroelectrics 147, 181±191 (1993). [72] J.-H. Chen, G.-H. Hsiue, C.-P. Hwang, and J.-L. Wu, Liq. Cryst. 19, 803±806 (1995).

10. Twist Grain Boundary Phases

351

[73] G.-H. Hsiue, C.-P. Hwang, J.-H. Chen, and R.-C. Chang, Liq. Cryst. 20, 45± 57 (1996). [74] S.-L. Wu and W.-J. Hsieh, Liq. Cryst. 21, 783±790 (1996). [75] W.-J. Hsieh and S.-L. Wu, Mol. Cryst. Liq. Cryst. 302, 253±269 (1997). [76] P. Styring, J.D. Vuijk, I. Nishiyama, A.J. Slaney, and J.W. Goodby, J. Mater. Chem. 3, 399±405 (1993). [77] A.C. Ribeiro, A. Dreyer, L. Oswald, J.F. Nicoud, A. Soldera, D. Guillon, and Y. Galerne, J. Phys. France II 4, 407±412 (1994). [78] W.M. Ho, H.N.C. Wong, L. Navailles, Ch. Destrade, and H.T. Nguyen, Tetrahedron 51, 7373±7388 (1995). [79] V. Vill, H.-W. Tunger, K. Hensen, H. Stegemeyer, and K. Diekmann, Liq. Cryst. 20, 449±452 (1996). [80] V. Vill, H.-W. Tunger, and M. von Minden, J. Mater. Chem. 6, 739±745 (1996). [81] I. Nishiyama and A. Yoshizawa, Liq. Cryst. 17, 555±569 (1994). [82] F. Hardouin, M.F. Achard, J.-I. Jin, J.-W. Shin, and Y.-K. Yun, J. Phys. France II 4, 627±643 (1994). [83] V. Faye, P. Barois, H.T. Nguyen, V. Laux, and N. Isaert, New J. Chem. 20, 283±286 (1996). [84] M.-H. Li, L. Detre, P. Cluzeau, N. Isaert, and H.-T. Nguyen, Liq. Cryst. 24, 347±359 (1998). [85] T. Seshadri and H.-J. Haupt, Chem. Commun. 1998, 735±736 (1998). [86] C.-S. Hsu and C.-H. Tsai, Liq. Cryst. 22, 669±677 (1997). [87] J. M. Gilli and M. KamayeÂ, Liq. Cryst. 12, 545±560 (1992). [88] L. Bata, K. Fodor-Csorba, J. Szabon, M.V. Kozlovsky, and S. Holly, Ferroelectrics 122, 149±158 (1991). [89] M.V. Kozlovsky, K. Fodor-Csorba, L. Bata, and V.P. Shibaev, European Polym. J. 28, 901±905 (1992). [90] E. I. Demikhov and M.V. Kozlovsky, Liq. Cryst. 18, 911±914 (1995). [91] M. Kozlovsky and E. Demikhov, Mol. Cryst. Liq. Cryst. 282, 11±16 (1996). [92] M.V. Kozlovsky, W. Haase, A. Stakhanov, and V.P. Shibaev, Mol. Cryst. Liq. Cryst. 321, 177±187 (1998). [93] M.V. Kozlovsky, V.P. Shibaev, A.I. Stakhanov, T. Weyrauch, and W. Haase, Liq. Cryst. 24, 759±767 (1998). [94] S.K. Prasad, V.N. Raja, G.G. Nair, and J.W. Goodby, Mol. Cryst. Liq. Cryst. 250, 239±245 (1994). [95] C.J. Booth, J.W. Goodby, J.P. Hardy, and K.J. Toyne, Liq. Cryst. 16, 43±51 (1994). [96] Y. Sah, Mol. Cryst. Liq. Cryst. 302, 207±213 (1997). [97] N. Kramarenko, V. Kulishov, L. Kutulya, and N. Shkolnikova, SPIE 2795, 83±91 (1996). [98] N. Kramarenko, V. Kulishov, L. Kutulya, V.P. Seminozhenko, A.S. Tolochko, and N.I. Shkolnikova, Kristallogra®ya 41, 1082±1086 (1996); Crystallog. Rep. 41, 1029±1033 (1996). [99] N.L. Kramarenko, V.I. Kulishov, L.A. Kutulya, G.P. Semenkova, V.P. Seminozhenko, and N.I. Shkolnikova, Liq. Cryst. 22, 535±541 (1997). [100] R.A. Lewthwaite, J.W. Goodby, and K.J. Toyne, Liq. Cryst. 16, 299±313 (1994).

352

H.-S. Kitzerow

[101] J.-H. Chen, R.-C. Chang, G.-H. Hsiue, F.-W. Guu, and S.-L. Wu, Liq. Cryst. 18, 291±301 (1995). [102] M. Young, G. Pitsi, M.-H. Li, H.-T. Nguyen, P. JameÂe, G. Sigaud, and J. Thoen, Liq. Cryst. 25, 387±391 (1998). [103] R. Shao, J. Pang, N.A. Clark, J.A. Rego, and D.M. Walba, Ferroelectrics 147, 255±262 (1993). [104] M. Werth, H.T. Nguyen, C. Destrade, and N. Isaert, Liq. Cryst. 17, 863±877 (1994). [105] I. Dierking, F. Giesselmann, J. Kusserow, and P. Zugenmaier, Liq. Cryst. 17, 243±261 (1994). [106] B.D. Terris, R.J. Twieg, C. Nguyen, G. Sigaud, and H.T. Nguyen, Europhys. Lett. 19, 85±90 (1992). [107] A. Borwitzky, G. Gesekus, and V. Vill, Proc. SPIE±Int. Soc. Opt. Eng. 3319, 105±108 (1998). [108] F. Grandjean, C. R. Acad. Sci. Paris 172, 71 (1921). [109] R. Cano, Bull. Soc. France Miner. Cristallogr. 90, 333 (1967). [110] N. Isaert, L. Navailles, P. Barois, and H.T. Nguyen, J. Phys. France II 4, 1501±1518 (1994). [111] I. Dierking, F. Giesselmann, and P. Zugenmaier, Liq. Cryst. 17, 17±22 (1994). [112] J.S. Patel, S.D. Lee, S.W. Suh, and J.W. Goodby, Liq. Cryst. 13, 313±317 (1993). [113] P.E. Cladis, A.J. Slaney, J.W. Goodby, and H.R. Brand, Phys. Rev. Lett. 72, 226±229 (1994). [114] Y. Galerne, J. Phys. France II 4, 1699±1711 (1994). [115] V. Vill and H.-W. Tunger, J. Chem. Soc., Chem. Commun. 1995, 1047 (1995). [116] V. Vill, H.-W. Tunger, and D. Peters, Liq. Cryst. 20, 547±552 (1996). [117] C. Lesser, Diploma-Thesis, Technical University Berlin, 1997. [118] G.S. Iannacchione and C.W. Garland, Phys. Rev. E 58, 595±601 (1998). [119] N. Andal and G.S. Ranganath, J. Phys. II 5, 1193±1207 (1995). [120] G. Joly, N. Isaert, H.T. Nguyen, and P. Barois, Ferroelectrics 179, 51±59 (1996). [121] K.A. Suresh, Mol. Cryst. Liq. Cryst, 302, 235±246 (1997). [122] A. Chanishvili, G. Chilaya, M. Neundorf, G. Pelzl, and G. Petriashvili, Cryst. Res. Technol. 31, 679±683 (1996). [123] J. Toner, Phys. Rev. B 43, 8289±8296 (1991). [124] J. Toner, Phys. Rev. B 46, 5715±5716 (1992). [125] K.J. Ihn, J.A.N. Zasadzinski, R. Pindak, A.J. Slaney, and J. Goodby, Science 258, 275±278 (1992). [126] C. Carboni, H.F. Gleeson, J.W. Goodby, and A.J. Slaney, Liq. Cryst. 14, 1991±2000 (1993). [127] S.K. Prasad, G.G. Nair, S. Chandrasekhar, and J.W. Goodby, Mol. Cryst. Liq. Cryst. 260, 387±394 (1995). [128] A. Anakkar, A. Daoudi, J.-M. Buisine, N. Isaert, F. Bougrioua, and H.T. Nguyen, Liq. Cryst. 20, 411±415 (1996). [129] N.A. Clark and S.T. Lagerwall, in: Ferroelectric Liquid Crystals: Principles, Properties and Applications Chap. I, (edited by J.W. Goodby, R. Blinc, N.A. Clark, S.T. Lagerwall, M.A. Osipov, S.A. Pikin, T. Sakurai, K. Yoshino, and B. eksÏ) Gordon & Breach, Philadelphia, 1991. [130] I. Dozov and G. Durand, Europhys. Lett. 28, 25±30 (1994).

10. Twist Grain Boundary Phases [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163]

353

I. Dozov, Phys. Rev. Lett. 74, 4245±4248 (1995). Y. Galerne, Phys. Rev. Lett. 72, 1299 (1994). L. Navailles, P. Barois, and H.T. Nguyen, Phys. Rev. Lett. 72, 1300 (1994). C.W. Garland, Mol. Cryst. Liq. Cryst. 288, 25±31 (1996). H.-J. Hug, A. Moser, O. Fritz, B. Stiefel, and I. Parashikov, Low Temperature Scanning Force Microscopy in: Forces in Scanning Probe Methods, NATOASI E (edited by H.-J. GuÈntherodt) Vol. 286, p. 35, 1995. A. Moser, H.-J. Hug, I. Parashikov, B. Stiefel, O. Fritz, H. Thomas, A. Barato¨, and H.-J. GuÈntherodt, Phys. Rev. Lett. 74, 1847 (1995). B.D. Yano¨, A.A. Ruether, and P.J. Collings, Liq. Cryst. 14, 1793 (1993). M. Petit, M. Nobili, and P. Barois, Eur. Phys. J. B 6, 341±345 (1998). M. Petit, P. Barois, and H.T. Nguyen, Europhys. Lett. 36, 185±190 (1996). P. Barois, M. Nobili, and M Petit, Mol. Cryst. Liq. Cryst. 302, 215±221 (1997). H.-S. Kitzerow, A.J. Slaney, and J.W. Goodby, Ferroelectrics 179, 61±80 (1996). C. Destrade, S. Payan, P. Cluzeau, and H.T. Nguyen, Liq. Cryst. 17, 291±297 (1994). H. Xu, Y.P. Panarin, J.K. Vij, A.J. Seed, M. Hird, and J.W. Goodby, J. Phys.: Condens. Matter 7, 7443±52 (1995). F. Bourgrioua, N. Isaert, C. Legrand, A. Bouchta, P. Barois, and H.T. Nguyen, Ferroelectrics 180, 35±48 (1996). S. Garo¨ and R.B. Meyer, Phys. Rev. Lett. 38, 848 (1977). J.L. Figueirinhas, A. Ferraz, A.C. Ribeiro, F. Noack, and H.T. Nguyen, J. Phys. France II 7, 79±91 (1997). P.L. Finn and P.E. Cladis, Mol. Cryst. Liq. Cryst. 84, 159±192 (1982). J.W. Goodby, D.A. Dunmur, and P.J. Collings, Liq. Cryst. 19, 703±709 (1995). Z. Kutnjak, C.W. Garland, C.G. Schatz, P.J. Collings, C.J. Booth, and J.W. Goodby, Phys. Rev. E 53, 4955±4963 (1996). E. Demikhov and H. Stegemeyer, Liq. Cryst. 10, 869±873 (1991). H. Stegemeyer, M. Schumacher, and E. Demikhov, Liq. Cryst. 15, 933±937 (1993). E. Demikhov, H. Stegemeyer, and V. Tsukruk, Phys. Rev. A 46, 4879 (1992). B. Pansu, M.H. Li, and H.T. Nguyen, J. Phys. France II 7, 751±763 (1997). M.-H. Li, H.-T. Nguyen, and G. Sigaud, Liq. Cryst. 20, 361±365 (1996). B. Pansu, M-.H. Li, and H.T. Nguyen, Eur. Phys. J. B 2, 143±146 (1998). A.-M. Levelut, C. Germain, P. Keller, L. LieÂbert, and J. Billard, J. Phys. France 44, 623 (1983). A.D.L. Chandani, E. Gorecka, Y. Ouchi, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 28, L-1265 (1989). Y. Galerne and L. LieÂbert, Phys. Rev. Lett. 64, 906 (1990). G. Heppke, P. Kleineberg, D. LoÈtzsch, S. Mery, and R. Shashidar, Mol. Cryst. Liq. Cryst. 231, 257 (1993). J.S. Patel, Phys. Rev. E 49, R3594±97 (1994). R.B. Meyer, Phys. Rev. Lett. 22, 918 (1969). T. Niori, T. Sekine, J. Watanabe, T. Furukawa, and H. Takezoe, J. Mater. Chem. 6, 1231±1233 (1996). D.R. Link, G. Natale, R. Shao, J.E. Maclennan, N.A. Clark, E. KoÈrblova, and D.M. Walba, Science 278, 1924 (1997).

354

H.-S. Kitzerow

[164] G. Heppke and D. Moro, Science 279, 1872±1873 (1998). [165] Y. Hatwalne, S. Ramaswamy, and J. Toner, Phys. Rev. Lett. 70, 2090±2093 (1993). [166] G. Yan and T.C. Lubensky, J. Phys. France II 7, 1023±1034 (1997). [167] R.D. Kamien and T.C. Lubensky, J. Phys. France II 7, 157±163 (1997). [168] T.C. Lubensky, T. Tokihiro, and S.R. Renn, Phys. Rev. A 43, 5449±5462 (1991). [169] I. Dierking and S.T. Lagerwall, Liq. Cryst. 26, 83±95 (1999).

11

Columnar Liquid Crystals Harald Bock

11.1

Introduction

Let us de®ne liquid crystals (as opposed to soft or plastic crystals and isotropic liquids) as states of condensed matter that are anisotropic and have positional order in less than three dimensions. We thus obtain three general classes of liquid crystals, two of which have already been dealt with in previous chapters: (1) anisotropic liquids without positional order, called nematic liquid crystals; (2) phases with positional order in one dimension (layers of twodimensional liquids), called smectic liquid crystals; and (3) phases with positional order in two dimensions (strings of onedimensional liquids), called columnar liquid crystals. The latter are formed by disk-like molecules (Figure 11.1). In columnar liquid crystals, the position of the molecule is thus ®xed with respect to the plane perpendicular to the column axis, whilst the molecular packing along the column axis is irregular, and the position of the molecule along that axis is not de®ned with respect to its neighbors in adjacent columns. How may columnar phases are chiral? We may consider: (a) chirality within the column; and (b) chirality of the lattice of columns. In both cases, the loss of mirror symmetry may be caused by a helix: either the orientation of the molecular director or the position of the molecules may spiral within the column, or the column lattice is helically distorted. Within the column, mirror symmetry may also be lost by the introduction of tilt and polarization. Similarly to a surface-stabilized (thus nonhelical) smectic C phase of a chiral material, a nonhelical columnar phase of chiral molecules may itself be termed chiral if the optical axis of the molecules is inclined with respect to the column axis: the vector of the tilt-induced po-

355

356

H. Bock

Figure 11.1. Molecular cores and chains mentioned in this chapter. Cores: a ˆ triphenylene, b ˆ dibenzopyrene, c ˆ hexahelicenetetrone, d ˆ phthalocyanine, e ˆ a-glucose, f ˆ b-glucose, g ˆ tricycloquinazoline, h ˆ truxene, i and j ˆ phenanthrene, k ˆ pyrene, l ˆ dibenzoperylene, m ˆ dibutyldibenzoperylene, n ˆ metal bis-diphenylpropanedionate. Chains: 3-methylalkanoate, O-alkyllactate, O-(1-methylalkyl)-glycolate, alkanoate, 3,7-dimethyloctyloxy, alkylthio, alkoxy, 3-methoxy-4-(1-methyl-alkoxy)-benzoate.

larization, the column axis, and the optical axis form a non mirror-symmetric arrangement (Figure 11.2) [1]. The chirality of the columnar phase may manifest itself, as in other liquid crystal phases, in a pronounced circular dichroism, in an ampli®ed speci®c rotation, in a piezoelectric response, and in electro-optical switching e¨ects. The columnar organization of concentrated DNA is one of the most prominent examples of the occurrence of liquid crystal phases in chiral natural systems.

11. Columnar Liquid Crystals

(a)

357

(b)

Figure 11.2. The origin of pyroelectricity in tilted chiral columnar (and smectic) phases; side view on disks (or rods); Y ˆ polar group; the column axis is vertical, the optical axis is tilted, and the vector of the macroscopic dipole is perpendicular to the plane of the paper. (Reprinted with permission from J. Mater. Chem. 5, 417 ( 1995, The Royal Society of Chemistry [1].)

11.2

Nonhelical (Polar) Chiral Structures

Most columnar phases are formed by more or less disk-shaped molecules whose two faces are either identical or not su½ciently di¨erent to induce polar packing (heads up, tails down) with respect to the column axis. If, in addition, the optical axis of the molecules is on average parallel to the column axis, then the phase has mirror symmetry as long as no helix is present. In most cases we can assume that the disks are essentially free to rotate around their short optical axis. But, as in smectic phases, the optical axis does not have to coincide with the column axis. Such tilted phases, which may be considered the columnar analogues of the smectic C phase(s), show a tilt-induced dipole moment if made up by chiral molecules and are thus pyroelectric, at least locally. The occurrence of a net dipole moment in tilted columnar (and smectic) phases can be most conveniently visualized and explained with a schematic molecule that consists of a rigid disk- (or rod-)shaped core, elongated ¯exible chains attached in two (or one) long direction(s), and chiral polar groups at the chain±core junctions (Figure 11.2). If the core and chains are of the same thickness, the optical axis of the molecules will tend to align along the column axis (or the layer normal). But if the core and chain are of di¨erent thicknesses, the formation of a kink between core and chain is favored. If the (e.g., acyloxy) chains are thicker than the (e.g., condensed aromatic) core, space is more e½ciently used and the void between cores is reduced, if the cores (but not the chains) are tilted toward the column axis (or the layer normal). Thus the optical axis tilts, and a kink forms between the core and

358

H. Bock

chain. This kink ®xes the orientation of the chiral polar groups that are positioned at the core±chain junction, which thus lose the rotational freedom they had in the nontilted phase. The dipoles thus ®xed at opposite sides of the molecule will have opposite components parallel to the column (or layer) axis, so no macroscopic dipole along that axis is created. However, the components perpendicular to that axis do not compensate each other, but a net dipole is created which is parallel to the axis about which the molecule is tilted. In smectic C phases, the order is only one-dimensional and only a few phases, that di¨er from each other in their pattern of tilt directions in different layers, have been found (SmC, SmCanti , SmCg , . . .). In columnar phases, on the other hand, the order is two-dimensional, and many phases with variously complex patterns of tilt direction in di¨erent columns are possible and have been reported. Whereas columns of nontilted disks generally form a hexagonal lattice, tilted columns are slightly elliptic and favor less symmetric rectangular and oblique lattices. The discovery of columnar phases in 1977 [2] was soon followed in 1980 by the ®rst studies of columnar phases of triphenylene-2,3,6,7,10,11-hexayl (S)-3-methylnonanoate that exhibits pyroelectric structures, as con®rmed by X-ray investigations [3]±[5]. Most interestingly, the chirality in this material manifests itself in beautiful spiraling defect structures (Figure 11.3). Free droplets on glass slides show spirals that are all left-handed in one enantiomer and all right-handed in the other. These may be caused by surface e¨ects or by a long-range helical structure of the column lattice (screw disclinations of one of the two handednesses may be preferred), although no such helix was reported from the X-ray investigations. The X-ray investigations indicated the presence of a helical rotation of the molecules only within the plane of the molecular core, but no helix of the tilt direction. A helical rotation of the tilt direction can also be excluded from the optical texture between crossed polarizers, where the macroscopic tilt can be detected as the inclination of the direction of maximum extinction versus the directions of the polarizers (Figure 11.4) [6]. In the chiral polar columnar phase of dibenzo[e,l ]pyrene-1,2,5,6,8,9,12,13octayl (S)-O-alkyllactates, not only can the tilt be observed by polarization microscopy, but a macroscopic dipole can also be measured by ferroelectric polarization reversal, thus indicating a chiral pyroelectric structure free of a tilt or dipole helix [7].

11.3 11.3.1

Helical Structures Helicity and Chiral Polar Structures

If a continuous twist is introduced around an axis perpendicular to the polar axis of a pyroelectric structure, the overall dipole vanishes within one full rotation. Consequently, structures such as the helical smectic C phase and its

11. Columnar Liquid Crystals

359

Figure 11.3. Chiral microscopic textures between crossed polarizers of the …‡† and …ÿ† enantiomers ((a) and (b), respectively) of triphenylenehexayl 3-methylnonanoate: (a1) and (b1): enantiomorphic opposite points in the high-temperature phase; (a1,2) and (b1,2): opposite points and spirals at the phase transition (schematic); (a2) and (b2): enantiomorphic spirals in the low-temperature phase. (Reprinted with permission from Nature 298, 46 ( 1982, Macmillan Magazines [4].)

360

H. Bock

Figure 11.4. Typical inclined extinction directions in columnar ¯ower-like domains between horizontal and vertical crossed polarizers and its interpretation assuming tilted disks in curled columns; extinction occurs in diagonal directions where the disks are parallel to a polarizer. (Reprinted with permission from Ferroelectrics 179, 203 ( 1996, Gordon and Breach [6].)

columnar analogues, tilted chiral columnar phases with a helix around the column axis, are not polar. The application of an electrical ®eld tends to distort the helix in such cases because the appearance of a macroscopic dipole is favored in response to the ®eld. On the other hand, if a helix is introduced around an axis which is not perpendicular to the dipole, an overall dipole is maintained. Therefore, if the column lattice (or layer structure) is screwed along a direction in the lattice (or layer) plane, the tilt-induced overall dipole is not necessarily lost.

11.3.2

Intracolumnar Helix Along the Column Axis

Three di¨erent helices may exist along the column axis: (a) the orientation of the molecule within the plane perpendicular to the column axis may turn; (b) the centers of gravity of the molecules may form a screw; and (c) the direction of tilt may rotate in tilted phases (Figure 11.5) [8]. All three may be coupled: if the molecules are not centrosymmetric, a rotation of their orientation will in most cases be accompanied by a shift of their center of mass; and if the center of mass rotates, tilting is favored. A helix of the molecular orientation has been observed in nontilted phases of achiral centrosymmetric molecules such as alkoxy [9] and alylthio derivatives of triphenylene [10], [11]. In one of the columnar phases of 2,3,6,7,10,11-hexakis-(hexylthio)-triphenylene, not only do the molecules form helices within the column, but also a regular pattern of right- and lefthanded helices is formed, where two-thirds of the columns have helices of one handedness and form circles around those with the other handedness (Figure 11.6). It has to be noted though that this phase is better described as soft crystalline than as liquid crystalline, as the positions of molecules in the

11. Columnar Liquid Crystals

361

Figure 11.5. Three fundamental helical arrangements within the column. (Reprinted with permission from Chem. Eur. J. 1, 171 ( 1995, Wiley-VCH [8].)

…b† …a†

Figure 11.6. Proposed structure of the three column superlattice of the helical phase of hexakishexylthio-triphenylene. (Reprinted with permission from J. Physique 50, 461 ( 1989, edp-sciences [10].)

362

H. Bock

Figure 11.7. (1) Column-forming hexahelicene; (2) column of disk-shaped molecules; and (3) column of helical molecules. (Reprinted with permission from J. Am. Chem. Soc. 120, 264 ( 1998, American Chemical Society [13].)

neighboring columns are registered. The helices are incommensurate, the number of molecules per full turn being 7.92, and a minimum correlation length of about 220 molecules (A 30 turns) was de®ned. To account for the lack of molecular chirality, it must be assumed that the samples consist of di¨erent (micro-)domains, in some of which left-handed and in others righthanded columns dominate. A highly organized (soft crystalline) helical columnar phase is also formed by enantiomerically pure 6,8,9,11-tetrakis-(dodecyloxy)-hexahelicene1,4,13,16-tetrone, where the molecular helix is basically extended into a column helix [12], [13] (Figure 11.7). The degree of organization along the column axis is impressively high, with the substance forming extended ¯exible ®bers, consisting of lamellae two to three columns high. A strong circular dichroism is found in concentrated solutions, and, as a remarkable sign of the ampli®cation of the molecular chirality by aggregation, a speci®c rotation ‰aŠD of 170,000 in the pure state, 280 times that of a dilute solution. Apart from the triphenylene mentioned in Section 11.2, one other case has been reported where a chiral columnar phase structure manifests itself in an asymmetric texture: 2,3,9,10,16,17,23,24-octakis-(S-3,7-dimethyloctyloxy)phthalocyanine exhibits two columnar phases above room temperature [8], [14]. The one at the higher temperature has a rectangular column lattice, the other a hexagonal one. When cooled quickly from the isotropic liquid, lefthanded spirals appear in the ¯ower-like texture of the highly ordered room temperature phase (Figure 11.8). X-ray and circular dichroism measureÊ (A 16 molecules). ments indicate a helical superstructure with a pitch of 55 A It seems very probable that in this case the molecules are weakly tilted and that the tilt direction spirals around the column axis, especially because the phase at higher temperatures is rectangular and therefore most probably tilted as well. A similar short-pitch helix of the tilt direction has been inferred from the electro-optical behavior of a switchable monotropic pyrene derivative (see

11. Columnar Liquid Crystals

363

Figure 11.8. Spiral texture of hexakis-(3,7-dimethyloctyloxy)-phthalocyanine observed between crossed polarizers after quickly cooling from the isotropic phase to room temperature. (Reprinted with permission from Chem. Commun. 1993, 1120 ( 1993, Royal Society of Chemistry [14].)

below). Some of the simplest chiral columnar liquid crystals are the pentaesters of a- and b-glucose, which form long-lived monotropic phases around room temperature [15]. Their X-ray and circular dichroism behavior is very dependent on thermal history. In annealed samples, a short-pitch helix along the column axis seems to be present at least in the more pronouncedly asymmetric a anomers. In a switchable rectangular columnar phase of a family of metal b-diketonates with chiral lactate chains, recent dielectric studies indicated the existence of a Goldstone mode, which necessitates a helical structure [16].

11.3.3

Rotation of the Column Lattice Along the Column Axis

In the same manner as the individual strands in a rope are twisted around each other, one can imagine a phase where several columns spiral around each other or around a straight column in the middle. Because such a rotation requires a single center of rotation, such a structure would not be periodic per se. But an in®nite number of such cords may be packed in a hexagonal lattice. Each cord may be made up of several inner and outer shells of columns. Kamien and Nelson [17] have constructed a detailed model of such ``moireÂ'' structures and elucidate stacked hexagonal arrays of screw disclinations within these structures, as shown in Figure 11.9.

364

H. Bock Figure 11.9. The moire state. The thick tubes running in the z-direction are columns, while the dark lines are stacked honeycomb arrays of screw dislocations. The intersection of these columns, with any constant z cross-section away from the hexagonal defect arrays, has the topology of a triangular lattice. (Reprinted with permission from Phys. Rev. E 53, 650 ( 1996, American Physical Society.)

11.3.4

Helix Perpendicular to the Column Axis

The situation is simpler if the column lattice is helically distorted perpendicular to the column axes: blocks of parallel columns are stacked on top of each other with a ®nite angle between the columns of adjacent blocks (Figure 11.10) [18]. The blocks are thus separated by planar tilt grain boundaries, and parallel linear screw disclinations lie within these boundary planes (Figure 11.11). This structure is perfectly analogous to the twist grain boundary phases of chiral smectic liquid crystals. No thermotropic columnar phase with a helical lattice distortion of either of the two above types has yet been reported, but concentrated solutions of DNA and poly-g-benzyl-L-glutamate show undulations and defects that indicate that such helical defect structures exist in these lyotropic systems (see section 11.5).

11. Columnar Liquid Crystals

365

Figure 11.10. Blocks of hexagonally ordered columns are helically stacked to form a tilt grain boundary phase. (Reprinted with permission from J. Physique 47, 1813 ( 1986, edp-sciences [18].)

Figure 11.11. A single screw disclination in a tilt grain boundary phase. The dark horizontal line is the screw disclination. (Reprinted with permission from Phys. Rev. E 53, 650 ( 1996, American Physical Society [17].)

11.4

Electro-Optical Properties

Whilst tilted smectic phases of chiral materials are in general electrooptically switchable, many chiral tilted columnar systems show no optical e¨ect whatsoever when an applied voltage is reversed (Figure 11.12). This resilience to the reorientation of dipoles by an electrical ®eld may in some cases be due to the more crystalline than liquid crystalline nature of the systems, but even in the quite conventional high-temperature tilted phase of the ®rst chiral material (triphenylene-2,3,6,7,10,11-hexayl (S)-3-

366

H. Bock

Figure 11.12. Electro-optical switching of tilted columnar phases. (Reprinted with permission from J. Mater. Chem. 5, 417 ( 1995, Royal Society of Chemistry [1].)

methylnonanoate), no switching can be induced [19]. With a few exceptions, all known switchable columnar materials bear O-alkyl-lactic acid chains that create strong tilt-induced dipoles [20]. Four di¨erent basic behaviors have been observed: (a) relatively small induced tilt angles (a few degrees at tens of V/mm) that are proportional to the applied electrical ®eld, short ®eld-independent switching times (tens of ms) and no hysteretic behavior; (b) larger ®eld-proportional switching angles at lower voltages that may suddenly increase and then saturate with increasing voltage, with ®eldindependent speed at low ®elds and ®eld-dependent speed above the saturation voltage; (c) bistable hysteretic switching between two discrete orientations of opposite tilt with switching speeds proportional to the ®eld; and (d) bistable hysteretic switching above and below a threshold ®eld, with a smaller discrete switching angle below and a larger one above the threshold and extremely ®eld- and temperature-dependent switching times. Case (a) may be interpreted as an electroclinic e¨ect similar to the one observed in nontilted smectic phases: tilt is induced because the inclination of disks with chiral polar side chains generates a dipole that counteracts the ®eld. This e¨ect is observed with triphenylene-2,3,6,7,10,11-hexayl, tricycloquinazoline-2,3,7,8,12,13-hexayl, and truxene-2,3,7,8,12,13-hexayl O-alkyllactates [19], [21]. For (b), two models are plausible: One may assume either an antiferroelectric pattern of tilted columns for which small ®elds cause a deviation of the tilt directions and for which a larger threshold ®eld changes the pattern of tilt directions into a ferroelectric structure. Alternatively, a helical tilt may be distorted at lower ®elds and unwound at higher ®elds. As the helix pitch in columnar phases is usually much shorter than the wavelength of visible light, no selective re¯ection of light is expected and one may best distinguish between the two cases by X-ray or circular dichroism measurements. Only

11. Columnar Liquid Crystals

367

Figure 11.13. Two antiferroelectric column lattices compatible with the X-ray diffraction patterns of a chiral phenanthrene-hexayl O-sec-alkyl-glycolate. (Reprinted with permission from Liq. Cryst. 24, 157 ( 1998, Taylor & Francis [23].)

for a single compound has the lattice structure been elucidated by X-ray scattering. The observed switching times with di¨erent materials vary greatly (tens of ms up to 1 second), which indicates considerable di¨erences in the mechanisms. Large-angle ®eld-proportional switching has been observed with phenanthrene-1,2,3,6,7,8-hexayl and -1,2,3,6,8-pentayl O-(1-methylalkyl)glycolates and O-alkyllactates, [21] with dibenzo[cd,lm]perylene-1,3,8,10-tetrayl and 2,9-dibutyl-dibenzo[cd,lm]perylene-1,3,8,10-tetrayl O-alkyllactates [22] and with some mixtures [1], [6]. X-ray di¨raction of the phenanthrenehexayl O-sec-alkyl-glycolate revealed an antiferroelectric structure with three columns per unit cell which are rotated by 120 relative to each other (Figure 11.13) [23]. The additional tilt saturation at high voltage has been measured in a pyrene-1,3,6,8-tetrayl Oalkyllactate (Figure 11.14) [24]. In this material, the lack of any hysteresis favors the helix unwinding model.

368

H. Bock

Figure 11.14. Optical tilt angle of a pyrene-tetrayl O-alkyl-lactate versus electric ®eld strength: …a† increasing ®eld strength; …G† decreasing ®eld strength, 26  C; triangular voltage, G12 (V mmÿ1 )/s. (Reprinted with permission from Liq. Cryst. 18, 707 ( 1995, Taylor & Francis [24].)

Cases (c) and (d) both represent truly ferroelectric behavior, which can be found in smectic C phases of chiral molecules only if the helix is suppressed by the boundary conditions at the liquid crystal glass interfaces. Because it is nearly impossible to align columnar phases uniformly with rubbed or otherwise anisotropic polymer surfaces (the switching is generally studied in polydomain or sheared samples), and because the ferroelectric behavior remains present in thick cells, surface induction of ferroelectricity can be excluded. It must be assumed that the two-dimensional column lattice suppresses helix formation in these materials. The ®eld-proportional speed behavior (c) is analogous to surface-stabilized smectic C; a pyroelectric pattern of tilt directions is reversed. This is observed with triphenylene-2,3,6,7,10,11-hexayl 3-methoxy-4-(1-methyl-heptyloxy)-benzoate [25]. The ®eld-induced transition between two switchable phases of di¨erent switching angle and di¨erent birefringence (case (d)) is observed in dibenzo[e,l ]pyrene-1,2,5,6,8,9,12,13-octayl O-alkyllactates. Similar to ferrielectric smectic Cg materials, a column lattice, where column dipoles partially compensate each other, changes at a threshold ®eld into a more polar structure. The extreme ®eld dependence of the speed (t ÿ1 z E 2 to E 5 ) is not known from smectics (t ÿ1 z E 1 to E 2 ) and is hard to understand. The strong variation of the speed with temperature may indicate a glass transition below the ferroelectric temperature domain (Figure 11.15) [7], [19]. The lattice distortion during switching also leads to an electromechanical (piezoelectric) response, i.e., to the movement of a cover glass on a switching sample. This e¨ect was found to be three orders of magnitude weaker than in chiral smectic liquid crystals [26].

11. Columnar Liquid Crystals

369

(a)

(b) Figure 11.15. Two fully switched stable states of the low ®eld phase of a ferroelectric dibenzopyrene-octayl O-alkyl-lactate, between crossed polarizers; no electrical ®eld applied. (Reprinted with permission from Liq. Cryst. 12, 697 ( 1992, Taylor & Francis [19].)

370

H. Bock

Figure 11.16. Highly concentrated liquid±crystalline phase of DNA observed by polarized light microscopy. A solution of 200 mg mlÿ1 DNA was deposited between slide and coverslip. The concentration was progressively increased by slow evaporation of the solvent and monitored between crossed polarizers. (a), (b) Highly birefringent germs of the hexagonal phase grow in the nearly extinguished planar cholesteric phase. They can be either (a) disk- or (b) diamond-shaped. (c)±(e) The hexagonal germs coalesce to form a homogeneous phase. Most of the domains articulate around disclination lines (L). Magni®cation 135 in (a), (e), 180 in (b), (d), 320 in (c). (Reprinted with permission from Nature 339, 724 ( 1989, Macmillan Magazines, [27].)

11.5

Occurrence in Nature [18], [27], [28]

Doubly stranded DNA is densely packed in cell nuclei, sperm heads, bacteria, and viruses, with an estimated concentration of up to 800 mg/ml. At lower concentrations in saline solutions, a cholesteric (chiral nematic) phase forms, but at hundreds of mg/ml, a typical columnar texture can be observed (Figure 11.16). PBLG (poly-g-benzyl-L-glutamate), also a double-helix polymer, exhibits similar textures. X-ray di¨raction and freeze-fracture investigations prove that the high concentration phase is indeed columnar, not smectic, with the molecular double helices packed into a hexagonal pattern (Figure 11.17). When the concentration is increased from the cholesteric to

11. Columnar Liquid Crystals

371

Figure 11.17. Arrangement of DNA or PBLG molecules in the columnar hexagonal phase. (Reprinted from J. Mol. Biol. 218, F. Livolant, Supramolecular organization of double-stranded DNA molecules in the columnar hexagonal liquid crystalline phase, pp. 165±181 ( 1991, by permission of Academic Press [28].)

the columnar region, a helical phase structure competes with a regular hexagonal lattice that excludes macroscopic continuous helices. This competition may lead to the helical moire and tilt grain boundary structures discussed by Kamien and Nelson [17]. Sinusoidal undulations of the optical axis, thin cholesteric layers between columnar domains, and nested-arc textures are all strong indications of a helical structure (Figures 11.18 and 11.19). The nested-arc texture shows the structure of the tilt grain boundary phase particularly well.

Figure 11.18. Hexagonal textures of PBLG. (a)±(c) Transformation from (a) undulating patterns to (c) herring-bone patterns when the polymer concentration is increased. (a) Undulating patterns showing numerous double defects. Dark lines follow the loci of maximum curvature of molecular orientations (120, crossed polarizers). (b) Loci of maximun curvature of molecules transform into walls of discontinuity and their path becomes sinuous (arrows). New undulations appear in each elongated domain …180†. (c) Herring-bone pattern …300†. (d) Other undulating patterns. The di¨erent domains are interrupted by walls (w). Illuminated regions draw either elongated rectangles (in the central domain) or elongated triangles (on the right) …300†. (e) Hexagonal patterns observed in free-surface drop. Molecular orientations are underlined by ®ne striae (s). Certain walls showing stair patterns are clearly observed (w) …300†. (f ), (g) Twist e¨ect observed between hexagonal domains in highly concentrated preparations. (f ) Beginning of a double spiral is created by the helical stacking of the domains in a free-surface drop (300, crossed polarizers, 2l=4). (g) Series of arcs produced by the same process, between slide and coverslip …330†. (Reprinted with permission from J. Physique 47, 1813 ( 1986, edp-sciences [18].)

Figure 11.19. Transition from the cholesteric to the hexagonal phase in PBLG (a), (b), (e), (f ), (g) and DNA (c), (d). (a) The transition from the vertical cholesteric phase (CH) to the hexagonal phase (H) occurs in the thickness of the preparation and certain cholesteric layers remain embedded in the hexagonal phase (arrow). The birefringence intensity is much stronger in the hexagonal phase …285†. (b) Sharp transition between the oblique cholesteric phase and the hexagonal phase which presents undulating patterns. The preparation is thinner than in (a) …550†. (c) Condensed DNA obtained in c conditions. The peripheral part of the aggregate presents a liquid crystalline organization which is ®rst cholesteric and planar (CH) and later transforms into a hexagonal phase (H) …150†. (d) Enlargement of the planar cholesteric domain in (d); hexagonal order appears along lines (arrows). (e) Planar hexagonal texture growing in the extinguished homeotropic phase around ‡p disclinations, near the interface with the isotropic phase (i) …315†. (f ) The planar hexagonal texture described in (e) progresses and forms elongated ribbons. The transition between the two regions is very sharp. Numerous ‡p and ÿp disclinations can be seen …290†. (g) Progressive transition from the homeotropic phase to the hexagonal phase. Inversion walls (I.W.) are observed in the homeotropic phase. …630†. (Reprinted with permission from J. Physique 47, 1813 ( 1986, edp-sciences [18].)

374

H. Bock

References [1] G. Scherowsky and X.H. Chen, J. Mater. Chem. 5, 417 (1995). [2] S. Chandrasekhar, B.K. Sadashiva, and K.A. Suresh, Pramana 9, 471 (1977). [3] C. Destrade, Nguyen Huu Tinh, J. MaltheÃte, and J. Jacques, J. Phys. Lett. 79A, 189 (1980). [4] J. MaltheÃte, J. Jacques, Nguyen Huu Tinh, and C. Destrade, Nature 298, 46 (1982). [5] A.M. Levelut, P. Oswald, A. Ghanem, and J. MaltheÃte, J. Physique 45, 745 (1984). [6] G. Heppke, D. KruÈerke, M. MuÈller, and H. Bock, Ferroelectrics 179, 203 (1996). [7] H. Bock and W. Helfrich, Liq. Cryst. 18, 387 (1995). [8] C.F. van Nostrum, A.W. Bosman, G.H. Gelinck, P.G. Schouten, J.M. Warman, A.P.M. Kentgens, M.A.C. Devillers, A. Meijerink, S.J. Picken, U. Sohling, A.-J. Schouten, and R.J.M. Nolte, Chem. Eur. J. 1, 171 (1995). [9] A.M. Levelut, J. Phys. Lett., Paris 40, L-81 (1979). [10] P.A. Heiney, E. Fontes, W.H. de Jeu, A. Riera, P. Carroll, and A.B. Smith, J. Phys. France 50, 461 (1989). [11] S.H.J. Idziak, P.A. Heiney, J.P. McCauley, P. Carroll, and A.B. Smith, Mol. Cryst. Liq. Cryst. 237, 271 (1993). [12] C. Nuckolls, T.J. Katz, and L. Castellanos, J. Am. Chem. Soc. 118, 3767 (1996). [13] A.J. Lovinger, C. Nuckolls, and T.J. Katz, J. Am. Chem. Soc. 120, 264 (1998). [14] C.F. van Nostrum, A.W. Bosman, G.H. Gelinck, S.J. Picken, P.G. Schouten, J.M. Warman, A.-J. Schouten, and R.J.M. Nolte, J. Chem. Soc., Chem. Commun. 1993, 1120. [15] N.L. Morris, R.G. Zimmermann, G.B. Jameson, A.W. Dalziel, P.M. Reuss, and R.G. Weiss, J. Am. Chem. Soc. 110, 2177 (1988). [16] B. Palacios, M. Rosario de la Fuente, M.A. Perez Jumbindo, R. Iglesias, J.L. Serrano, and T. Sierra, Liq. Cryst. 25, 481 (1998). [17] R.D. Kamien and D.R. Nelson, Phys. Rev. E 53, 650 (1996). [18] F. Livolant and Y. Bouligand, J. Physique 47, 1813 (1986). [19] H. Bock and W. Helfrich, Liq. Cryst. 12, 697 (1992). [20] S. Kobayashi, S. Ishibashi, and S. Tsuru, Mol. Cryst. Liq. Cryst. Lett. 7, 105 (1990). [21] H. Bock, unpublished results (1992±1995) [22] G. Scherowsky and X.H. Chen, Liq. Cryst. 17, 803 (1994). [23] G. Scherowsky, X.H. Chen, and A.-M. Levelut, Liq. Cryst. 24, 157 (1998). [24] H. Bock and W. Helfrich, Liq. Cryst. 18, 707 (1995). [25] G. Heppke, D. LoÈtzsch, M. MuÈller, and H. Sawade, 6th International Conference on Ferroelectric Liquid Crystals (FLC 97), ENST, Bretagne in Brest, France, July 20±24, 1997. [26] A. JaÂkli, M. MuÈller, D. KruÈerke, and G. Heppke, Liq. Cryst. 24, 467 (1998). [27] F. Livolant, A.M. Levelut, J. Doucet, and J.P. Benoit, Nature 339, 724 (1989). [28] F. Livolant, J. Mol. Biol. 218, 165 (1991).

12

Some Aspects of Polymer Dispersed and Polymer Stabilized Chiral Liquid Crystals Gregory P. Crawford, Daniel SvensÏek, and Slobodan ZÏumer

Systems consisting of a low molar mass liquid crystal and a polymer are currently of great interest with respect to applications and due to the intriguing ®nite size e¨ects. This chapter describes some aspects of liquid crystals embedded in dense polymer binders and low concentration polymer networks that modify the bulk liquid crystal phase, with added emphasis on chirality. The introduction to the phenomenological description is followed by the modeling of ®eld e¨ected chiral nematic droplets in polymer-dispersed liquid crystal systems. Next the orientational ordering induced by polymer networks is described, and ®nally the usefulness of these materials for directview re¯ection displays, bistable displays, and light valves is reviewed.

12.1

Introduction

The nature of con®ning surfaces on most materials (e.g., solids and liquids) has little in¯uence on the internal properties, and, for the most part, can be safely ignored even for small-scale samples. Liquid crystalline materials, on the other hand, are an exception to this rule. The con®nement of a liquid crystal material can modify ordering over macroscopic distances; in some cases, the bulk order of a liquid crystal material is determined in millimetersized samples. This unique ability to manipulate the bulk liquid crystal structure via properties of the boundary, along with the ability to modify and control that structure using applied electric and/or magnetic ®elds, is the main reason why liquid crystal materials are ideal for a host of electro-optical applications. In addition to the direct e¨ect of the surface in real liquid crystal samples, the e¨ects of the con®ning geometry are also important. The simplest example of a particular geometry is the liquid crystalline cell with the con®nement between parallel plates, separated by a few micrometers, which is widely used in ¯at panel liquid crystal display applications. In recent years there has been a move to study more complex con®ned liquid crystal systems with a large surface-to-volume ratio [1], [2]. Fundamentally, the e¨ect of geometry and surface-induced ordering on liquid crystalline phases 375

376

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.1. Schematic presentation of the operation of a PDLC shutter.

is intriguing and conceptually challenging. On the other hand, it is perhaps even more important that the applicability of these complex microcon®ned liquid crystals is gaining widespread attention for scattering, re¯ective, and bistable display applications. In the mid-1980s the usefulness of con®ning liquid crystals to spherical droplets became readily apparent [3]. This came almost a century after the ®rst identi®cation of the liquid crystal phase in 1887 by Reisser and over 80 years after the ®rst liquid crystal droplets were observed by Lemman in a viscous liquid medium. Figure 12.1 presents a simple schematic of the operation of a polymer-dispersed liquid crystal (PDLC) device [2], [3], which depicts the ®rst practical demonstration of liquid crystal droplets. A rigid polymer binder permanently supports the liquid crystal droplets. In the passive state (no applied voltage), the symmetry axes of the liquid crystal con®gurations within the droplets are randomly oriented. The droplets in Figure 12.1 show the well-known bipolar con®guration, which is most common in nematic liquid crystal and polymer systems. This randomly oriented droplet system scatters light because of the mismatch between the average index of refraction of the droplet and the polymer binder. In its active state when a voltage of su½cient magnitude is applied, the droplets will reorient their symmetry axes parallel to the applied ®eld direction for materials with a positive dielectric anisotropy. If care is taken to select a liquid crystal with an ordinary index of refraction n0 , that approximately matches that of the polymer …np †, the material is optically homogenous and is therefore transparent. There have been a number of reviews on conventional PDLCs in the literature [2], [4]. In the spirit of this book, we will focus our attention to liquid crystal polymer systems that utilize chiral liquid crystal materials. Con®ned chiral nematic (N*) liquid crystals have also been the subject of numerous investigations, just as their nematic counterpart described above

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

377

[5], [6], [7]. Consequently, the development of PDLCs has been extended to chiral liquid crystals. Crooker and Yang [9] and Kitzerow and Crooker [10] recognized the usefulness of con®ned chiral materials for re¯ective display applications. These papers, in the early 1990s, rejuvenated interest in these systems beyond the scope of applications, and stimulated intense basic research in the area of di¨erent con®ned chiral liquid crystals [7], [8], [11], [12], [13], [14], [15], [16]. During the same time-frame of the ®rst reports of chiral PDLCs, another polymer-based liquid crystal system was reported by Hikmet [17] (1990). In contrast to PDLCs, these systems contained only a small volume fraction of a highly cross-linked mesogenic polymer network dispersed in the liquid crystal. Shortly after the report by Hikmet on lowconcentration networks in nematic liquid crystals, Yang and coworkers [18] (1992) adapted these materials to chiral nematic liquid crystal systems. These polymer-stabilized chiral liquid crystals were immediately realized to be useful for normal and reverse mode light shutters and re¯ective displays with bistable memory [19]. In this chapter we will present selected aspects important for the understanding of polymer-dispersed and polymer-stabilized chiral nematic liquid crystals. In addition to a brief review of applications of chiral nematic± polymer composites, we have in particular selected two seldom-discussed topics: chiral nematic structures in spherical cavities and polymer-induced ordering in the liquid crystal. They must be taken as rather simple but instructive examples of ordering in these systems. The chapter is organized in the following way: it begins with a brief introduction to materials and with a phenomenological description of nematic and chiral nematic ordering, it then treats in detail ®eld-induced structural transition in chiral droplets embedded in a polymer matrix. The next section is devoted to polymer-induced pretransitional orientational ordering and to the network structure determination. Finally, several optical display applications are described.

12.2

Polymer-Liquid Crystal Dispersions

Polymer liquid crystal dispersions have been extensively reviewed elsewhere [2], [4], [19], therefore we will only provide a sample of the rich variety of polymer morphologies that are possible. These complex systems are developed using a phase separation process. Separations caused by free radical or photo-induced polymerization in a mixture of monomers and liquid crystal are the most common. Thermally induced phase separation is used with a solution of thermoplastic polymers and liquid crystals. Solvent-induced phase separation based on solvent evaporation is usually used with thermoplastics which decompose before melting. Morphologies of the polymer matrix on submicrometer scale are most directly observed by scanning electron microscopy (SEM) after removal of the liquid crystal. Figure 12.2 shows three SEM photographs that are radically di¨erent. Figure 12.2(a) shows a con-

378

G.P. Crawford, D. SvensÏek, and S. ZÏumer (a)

Figure 12.2. Polymer morphology recorded with scanning electron microscopy: (a) classical PDLC display material and (b) and (c) polymer network morphologies formed under di¨erent initial conditions (courtesy of Y.K. Fung).

(b)

(c)

ventional PDLC matrix where the polymer content was highÐapproximately 50% by weight. When only low concentrations of polymer are used, as shown in Figure 12.2(b) and (c), fascinating polymer network morphologies are formed that strongly depend on conditions under which they were phase separated. The structure shown in Figure 12.2(b) was formed under homeotropic alignment conditions in zero ®eld, while that of Figure 12.2(c) was formed under homogenous surface anchoring conditions in the presence of a strong electric ®eld. It is very di½cult to classify the morphologies because of their strong dependence on initial conditions, but it can be said that conventional PDLCs have droplets that are separated from each other, while the low-concentration polymer networks are a bicontinuous medium with the interconnected cavities.

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

379

Conventional chiral PDLCs are formed using a liquid crystal material with a negative dielectric anisotropy, doped with a chiral component [9], [13], [19]. A typical mixture is composed of the commercially available Merck material ZLI-2806, which has a negative dielectric anisotropy, and a chiral dopant such as CE2, also available from Merck. These materials are combined with a thermosetting polymer, polyvinyl butyral (PVB) in the proportions ZLI-2806 (33 wt.%), CE2 (20 wt.%), and PVB (47 wt.%). The solution is completely mixed in chloroform and decanted onto conducting substrates at room temperature. Chloroform evaporates, the sample is heated to 140  C, compressed with another conduction substrate, and allowed to cool. Micrometer spacers are typically used to control the cell gap between the two substrates. The droplet size is determined by the cooling rate [10]. This is only one example of a cholesteric PDLC formulation that re¯ects light at approximately 520 nm. The SEM photographs of these materials reveal isolated droplets, reminiscent of the photograph in Figure 12.2(a). This matrix promotes the planar anchoring of liquid crystal molecules and thus the formation of spherical and oblate chiral structures discussed in Section 12.4. The low-concentration networks, presented in Figure 12.2(b) and (c), are fabricated by mixing small concentrations of reactive monomer (0.5±5 wt.%) into a liquid crystal [17], [18], [19]. Due to the similarity of the molecular structure of the reactive monomer, it typically aligns within the liquid crystal con®guration through steric hindrance. Reactive monomers are typically acrylate based and therefore they are polymerized by ultraviolet (UV) light. Many di¨erent structures have been synthesized and used for polymerstabilized liquid crystals [19]. The polymer morphologies strongly depend on initial conditions prior to the UV polymerization, monomer concentration, and UV curing conditions [20], [21]. Studies by Dierking have revealed that mean pore size in the polymer networks strongly depends on the curing temperature [20] and UV curing conditions [21], and that this network is responsible for two di¨erent switching regions [22] upon application of an electric ®eld. A detailed description of the investigation of the network structure via pretransitional orientational ordering, induced by a polymer in an isotropic liquid crystal, is given in Section 12.7.

12.3

Description of Nematic and Chiral Nematic Ordering

In this section we brie¯y discuss the most important aspects of the phenomenological description of nematic and chiral nematic systems, needed for the understanding of our particular examples in Sections 12.4 and 12.5. More details on this subject can be found in the general references [25], [26], [27], [28], [29], [30], [39], and references therein. Unlike ordinary liquid, a nematic liquid crystal is macroscopically aniso-

380

G.P. Crawford, D. SvensÏek, and S. ZÏumer

tropic, meaning that it exhibits a long-range order of molecular orientation. On average its molecules (essentially being rod- or disk-like) point in a certain direction, which is represented by a unit vector called the director ~ n. In an undeformed homogeneous nematic, ~ n is constant throughout the sample. In an undeformed chiral nematic (also called cholesteric) however, the director rotates about a constant perpendicular helical axis as we go along it. Locally the cholesteric appears like the ordinary nematic, only over larger scales characterized by the pitch length (the length of one revolution), spiraling becomes manifest. The pitch length in the range of wavelengths of visible light is responsible for the remarkable optical properties of cholesterics (see [25], for instance).

12.3.1

The Order Parameter

To describe the distribution of molecular orientations, a second-rank symmetric traceless tensor is introduced as an order parameter [27, p. 168]: Qij ˆ hdi dj i ÿ 13 dij ;

…12:1†

where h i stands for the statistical average over all possible molecular orientations represented by the unit vectors ~ d. A properly weighted unit tensor has been subtracted so that the order parameter is zero in the isotropic phase. The order parameter (12.1) is nothing but a traceless quadrupole tensor of the distribution of molecular orientations. It is the ®rst nontrivial nonzero moment, too, since ÿ~ n represents the same orientation as ~ n (the director is a ``headless'' vector) and therefore the dipole moment vanishes. A symmetric traceless tensor contains ®ve independent scalar quantities, hence it follows that one needs that many scalars to describe nematic ordering in a complete manner. Most conveniently, the tensorial order parameter is viewed in its eigensystem 3 2 bÿs 7 6 3 7 6 ÿb ÿ s 7 6 …12:2† Qˆ6 7; 7 6 3 5 4 2s 3 the z-axis was chosen to coincide with the director ~ n. In (12.2), two quantities have been introduced, s being the scalar order parameter sˆ

3hcos 2 Qi ÿ 1 ; 2

cos Q ˆ ~ n
~ d;

…12:3†

and b giving the degree of biaxiality, i.e., nonuniformity of the distribution projected onto the plane perpendicular to the director, ranging from b ˆ ÿ…s ÿ 1† (reached when hdy2 i ˆ 0) to b ˆ s ÿ 1 (achieved for hdx2 i ˆ 0),

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

381

whereas b ˆ 0 corresponds to the uniaxial distribution, of course. The scalar order parameter s can be both positive or negative, as seen from (12.3). In the latter case, molecular axes align perpendicularly to the director, so they tend to lie in a plane, as opposed to orienting along a single direction when s > 0. Now the need of ®ve scalar parameters can be understood more clearly, namely, three of them are necessary to de®ne the eigensystem, with the remaining two specifying the degree of order (s) and biaxiality …b†.

12.3.2

The Free Energy

When dealing with systems at constant temperature, the free energy F is the proper thermodynamic potential minimized in equilibrium, provided that there is no work done on the system (dF U 0 is valid when approaching equilibrium). The (nonequilibrium) free energy density must be expressed as a functional of the order parameter pro®le. Then its minimum will correspond to the equilibrium order parameter ®eld. According to Landau, near the phase transition the free energy density f is expressed in powers of the order parameter and its derivatives. In the presence of electric and magnetic ®elds also their contribution to the free energy density must be taken into account. In the case of the nematic liquid crystal, the order parameter (12.1) is a tensor, so the free energy density expression must be composed of scalar invariants formed by Q, its spatial derivatives, and external ®eld vectors. Including terms of the fourth order, the part of the free energy density not depending on inhomogenities (the so-called bulk contribution) and external ®elds reads (see [28, p. 156], for instance): fb ˆ f0 ‡ 12 A…T ÿ T  † Tr Q 2 ÿ 13 B Tr Q 3 ‡ 14 C…Tr Q 2 † 2 ‡


; …12:4† where 2 Tr Q 4 ˆ …Tr Q 2 † 2 holds for Q's of the form (12.2), so only one of the two needs to be included in (12.4). The free energy density of the isotropic phase has been denoted f0 . The constants A and C are positive, while B may be either positive or negative. The cubic term is necessary, because Q and ÿQ describe di¨erent states, which results in a discontinuous phase transition. The temperature T  is the isotropic phase supercooling temperature. The part of the free energy density resulting from spatial variation of the order parameter (the so-called deformation terms) is expressed as [28, p. 156]: . . …2† …2† …2† fd ˆ L1 …`Q†..…`Q† ‡ L2 …`
Q†
…`
Q† ‡ L3 …`Q†..…`Q†y . . …2† …2† ‡ L…1† `
…`
Q† ‡ L5 …`…`
Q††..Q ‡ L6 …` 2 Q†..Q ‡ higher order terms; …n† Li . are

…12:5†

temperature-independent generalized elastic constants of the where nth order, .. denotes contraction to a scalar, and …qi Qjk †y ˆ qj Qik . In the ®rst row of (12.5) terms quadratic in Q containing ®rst derivatives are listed, in the second one those containing second derivatives have been collected,

382

G.P. Crawford, D. SvensÏek, and S. ZÏumer

either linear or quadratic in Q. All terms given by (12.5) are invariant against rotations as well as inversion of the coordinate system. To describe a chiral nematic phase, however, a term has to be added to (12.5) that is not invariant against inversion in order to re¯ect the symmetry of chiral structures. In the lowest order this gives [37]: …2†

fc ˆ L4 eijk Qil qk Qjl :

…12:6†

On inversion of the coordinate system fc changes sign, hence it is a pseudoscalar. It is this term what makes the director rotate in a helical manner and causes a nematic to become a cholesteric. Such a situation occurs if a substance consisting of chiral molecules is dissolved in an ordinary nematic sample. The same structure has been found with pure cholesterol esters as well, so this is where the name ``cholesteric'' comes from. With electric or magnetic ®elds present, additional energy contributions must be taken into account. To be more precise, the free energy of dipoles in external ®elds is to be considered, e.g., the free energy density due to magnetization in a magnetic ®eld … H0 ~0 † ˆ ÿm0 ~ H† ~
d H; ~ M… …12:7† f m …H 0

~ is the magnetic ®eld strength and M ~ magnetization, depending where H ~ In the lowest order, construction of invariants from the tensor linearly on H. order parameter and the ®eld vectors, not including any spatial derivatives, yields ~
Q
E ~ÿ fem ˆ ÿ12 e0 Xe E

1 ~
Q
B; ~ Xm B 2m0

…12:8†

where Xe and Xm are electric and magnetic susceptibility anisotropies, re~ For simplicity, terms con~ ˆ m0 H. spectively, to be speci®ed below, and B taining derivatives were not included in (12.8), because of this some possible energy contributions were not taken into account, like the energy of a dipole in an inhomogeneous ®eld and, on the other hand, the so-called orderelectric [44] and ¯exoelectric [25, p. 135] e¨ects, i.e., polarization as a result of inhomogenities of the order parameter, thus resulting in an energy contribution when in external ®eld. The total free energy density is the sum of individual contributions f ˆ fb ‡ fd ‡ fc ‡ fem ;

…12:9†

thus being a functional of the order parameter and its spatial derivatives ! qQ qQ 2 ; ;... : …12:10† f ˆ f Q; q~ r qr 2 As discussed above the free energy is minimal when the system reaches

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

383

equilibrium, or in other words, when the order parameter takes up its equilibrium pro®le. This requires that the variation of the free energy be zero … dF ˆ d f dV ˆ 0; …12:11† from which a set of Euler±Lagrange equations is obtained. In general, this set is di½cult to solve, therefore usually the equilibrium con®guration is obtained from a simpli®ed free energy. A homogeneous undeformed nematic exhibits a uniaxial structure. In this case, the order parameter (12.1) can be written in a simpler form as Qij ˆ s…ni nj ÿ 13 dij †;

…12:12†

where s is the scalar order parameter, de®ned in (12.3), measuring the degree of molecular alignment with the director ~ n, or, when expressed in the eigensystem 3 2 1 ÿ3 7 6 Q ˆ s4 …12:13† ÿ 13 5; 2 3

as before (12.2) ~ n points along the z-axis. Now the temperature-dependence of the scalar order parameter s can be studied, rewriting (12.4): f ˆ f0 ‡ 12 a…T ÿ T  †s 2 ÿ 13 bs 3 ‡ 14 cs 4 ‡


;

…12:14†

where a new set of temperature-independent constants has been introduced. For a typical liquid crystal like the 5CB it has been found that experimental results are best described with a ˆ 0:13
10 6 J/m 3 K, b ˆ 3:89
10 6 J/m 3 , c ˆ 3:92
10 6 J/m 3 , and T  ˆ 307 K. As soon as a deformation is present, however, s is no more constant in space, and, in general, the biaxiality b is no longer zero. This can be easily understood from a symmetry point of view; as soon as a deformation occurs a new direction in space is de®ned beside the director and, hence, there is no particular reason for the uniaxial state to have the lowest energy. To get an insight into the case when the degree of order s is inhomogeneous, it is instructive to consider a situation where, in the tensor (12.12), only left s is spatially dependent. Taking into account only the ®rst term in (12.5) the …2† deformation free energy density reduces to fd ˆ L…`s† 2 =2, with L1 written simply as L. After combining fd with the bulk free energy (keeping only the quadratic term in (12.14)) one ®nds that a variation of the order parameter s decays with a characteristic length s L : …12:15† xˆ a…T ÿ T  † This quantity is referred to as the nematic correlation length. For 5CB at the phase transition temperature it measures approximately 10 nm.

384

G.P. Crawford, D. SvensÏek, and S. ZÏumer

12.3.3

The Frank Elastic Theory

In the case of weak deformation, it is possible to consider b to be zero together with s constant, while keeping only spatial variation of the director. Then the deformation free energy (12.5) can be rewritten as the so-called Frank elastic energy [25]: n† 2 ‡ 12 K22 ‰~ n
…`  ~ n† ‡ q0 Š 2 ‡ 12 K33 ‰~ n  …`  ~ n†Š 2 fd ˆ 12 K11 …`
~ ÿ 12 K24 `
‰~ n…`
~ n† ‡ ~ n  …`  ~ n†Š ‡ K13 `
‰~ n…`
~ n†Š:

…12:16†

Frank elastic constants were introduced giving the energy cost of individual deformation modes: K11 for splayed, K22 for twisted, and K33 for bent director ®elds. The K24 and K13 terms can be converted to a surface free energy density contribution on the sample boundary, as indicated by writing them as a divergence. Only in the case of ®xed boundary conditions can they be …n† dropped. The elastic constants Kij can be expressed in terms of Li and s, thus they are functions of temperature [37]. It should be mentioned that cubic terms are required in (12.5) in order to have K11 0 K33 , a situation that is usually observed experimentally. In the twist term a part of fc (12.6) has …2† been included, with q0 K22 ˆ s 2 L4 . Apparently, the twist term is minimized when ~ n
…`  ~ n† ˆ ÿq0 , resulting in a spontaneously twisted director ®eld characteristic for cholesterics. The director rotates by p on the pitch length p ; …12:17† lc ˆ jq0 j typically ranging from 0.1 mm to 100 mm. The parameter q0 is referred to as the chirality. Finally, the electric and magnetic free energies (12.8) are expressed in terms of s and ~ n: 1 ~
~ ~
~ n† 2 ÿ 13 E 2 † ÿ Xm s……B n† 2 ÿ 13 B 2 †; fem ˆ ÿ12 e0 Xe s……E 2m0

…12:18†

where Xe and Xm are microscopic susceptibility anisotropies, i.e., di¨erences between susceptibilities along the molecular axis and susceptibilities in the 0 0 0 0 ÿ we? , Xm ˆ wmk ÿ wm? ). Multiplied by s perpendicular direction (Xe ˆ wek they represent di¨erences in susceptibility along the director and in perpendicular direction as observed macroscopically (for a cholesteric this is true only on small scales compared to the length lc (12.17)): wea ˆ wek ÿ we? ˆ Xe s;

wma ˆ wmk ÿ wm? ˆ Xm s:

…12:19†

It is worth mentioning that in (12.18) only those contributions have been retained that depend on the order parameter, whereas the order parameter 0 0 0 0 ‡ 2we? †E 2 and ÿ…1=2m0 † 13 …wmk ‡ 2wm? †B 2 , independent terms ÿ…e0 =2† 13 …wek respectively, have not appeared. This is due to the fact that invariants of the form E 2 and B 2 had not been included in (12.8).

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

385

Frequently we want to keep the calculations as simple as possible. In this case the so-called one-constant approximation is introduced, setting all elastic constants Kii in (12.16) equal and dropping the surface terms. In this way a simple expression for the elastic distortion energy is obtained fd ˆ 12 Kqi nj qi nj :

12.3.4

…12:20†

Anchoring and Characteristic Lengths

Until now we have been discussing solely the bulk properties of the nematic liquid crystal, paying no attention to what is going on at its boundaries, e.g., container walls, contact with air, etc. Generally, the value of the order parameter at the boundary is in¯uenced by properties of the bounding surface. It can favor di¨erent director arrangements, e.g., perpendicular (homeotropic), parallel orientation, etc., what is more, the degree of ordering can be suggested as well. These e¨ects are known as the surface anchoring. Usually anchoring is modeled by a short-range surface±nematic interaction, expressed simply in terms of a delta function [35], [36]: ~ r† ˆ 12 W0 …1 ÿ …~ n
~ k† 2 †d…~ r ÿ R†; fs …~

…12:21†

k the preferred direction. with W0 being the zenithal anchoring strength and ~ If ~ k is normal to the surface then (12.21) describes homeotropic anchoring, whereas degenerate planar anchoring is described by ~ r† ˆ 12 W0 …~ n
~ k† 2 d…~ r ÿ R†: fs …~

…12:22†

In general the azimuthal (in plane) anchoring must also be taken into account (see, for instance, [31]). Close to the con®ning surfaces on distances comparable to the nematic correlation length (12.15) the degree of order s and biaxiality b are also a¨ected by the presence of the interface [32]. To describe such e¨ects the anchoring free energy usually is expressed in terms of scalar invariants formed from the tensor order parameter Q and parameters characterizing the anchoring. To visualize this possibility a simple model [33] should be mentioned, favoring a (uniaxial) degree of order s0 as well as a surface director ~ k: fs ˆ 12 w0 Tr…Q ÿ Q0 † 2 ;

…12:23†

with w0 as a coupling constant Q0ij ˆ s0 …ki kj ÿ 13 dij †. Often we call for simplicity and try to neglect the free energy contribution of the anchoring. In this case two possibilities emerge; either we neglect the anchoring free energy, which results in no anchoring (or better, the weak anchoring limit), or, we consider it having an in®nite value as soon as the director deviates from the favored arrangement, implying strong anchoring with ®xed con®guration at the surface. The relevance of the anchoring is often estimated in terms of the so-called extrapolation length, de®ned as [25, p. 113]:

386

G.P. Crawford, D. SvensÏek, and S. ZÏumer

lˆ

K ; W0

…12:24†

where K is one of the elastic constants or their combination. Evidently, l, ranging from 100 nm to 100 mm, is a measure of the anchoring strength compared to the energy of elastic distortion. E¨ectively, strong anchoring corresponds to l f R, where R is a typical dimension over which elastic deformation takes place and, conversely, l g R means that the weak anchoring limit is a reasonable approximation. In the chiral nematic liquid crystal, however, the extrapolation length has to be compared to the pitch length lc (12.17) as well, in order to estimate the importance of anchoring. Similarly, the strength of external ®elds is usually characterized by corresponding lengths, where ®eld e¨ects prevail over the elastic resistance of the nematic against distortion. In this way, electric (xe ) and magnetic (xm ) coherence lengths can be de®ned [25, p. 123], saying that they are typical lengths, over which electrically or magnetically induced order is restored, if the director is in some place brought out of alignment with the ®eld s s K m0 K ; xm ˆ ; …12:25† xe ˆ a 2 e0 jwe jE jwma jB 2 where electric and magnetic susceptibility anisotropies have been de®ned in (12.19). Again, a small coherence length means that the interaction with the ®eld is strong, and vice versa. For a typical liquid crystal these lengths are about 1 mm in ®elds B ˆ 4 T or E ˆ 1 V/mm.

12.3.5

Defects in Nematics and Cholesterics

Let us ®rst try to give a useful de®nition of defects. Naively we can say that a defect is an irregularity in the order parameter ®eld, i.e., a discontinuity. This can happen in a single point, a line, or a plane, resulting in zero-, one-, and two-dimensional defects. Their fundamental properties depend on the order parameter, or more precisely, on its symmetry. We can say that defects are a ``registration mark'' of systems with broken symmetry. Of course, physically it is hardly possible to speak about any discontinuities, so there may be a problem with our initial de®nition of the defect. In a nematic, a discontinuity is present only as long as the director description is considered, with s kept ®xed; if this restriction is abandoned and changes of degree of order are allowed, a continuous solution is obtained [38]. Therefore a more general de®nition of a defect must be searched for. Figure 12.3 shows a point defect in two dimensions or a cross-section of a line defect in three dimensions present in the center, arising as a result of a homeotropic con®nement. If a loop is imagined around the defect and then traversed counterclockwise so as to return to the starting point, a so-called winding number n can be de®ned as a measure of the total angle the director is rotated by on this trip, n ˆ f=2p. Since the loop passes over a defectless structure only, the con-

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

387

Figure 12.3. A point defect as a result of a homeotropic boundary condition. The ®gure can be interpreted as a cross-section of a line defect, as well. A loop is placed around it to de®ne its winding number n. For the defect shown, the winding number is n ˆ 1.

tinuity of the director ®eld imposes that the angle of rotation must be an integer multiple of p, so n ˆ 0; G12; G1; G32; . . . :

…12:26†

The winding number does not depend on the size or actual shape of the loop but solely on the type of the defect encircled. Therefore the winding number n identi®es the defect completely and is sometimes referred to as the strength of the defect. Even if the singularity in the center (called the core of the defect) is somehow smeared (e.g., by melting, to be the case below) the winding number does not change. In fact, to determine the strength of the defect n we do not need any information whatsoever on what the central con®guration is. Therefore the core region is not of importance for the macroscopic description of the so-called topological defects, i.e., defects that cannot be converted to a defectless structure by means of any continuous transformation of the director ®eld [39]. Topologically speaking, all defects transformable into each other by continuous transformations are identical. This means that those which can be so transformed to a defectless structure, are not defects in the topological sense. From now on only topological defects will be considered. Usually topological defects are energetically stable, although they do not correspond to states of the lowest free energy [27]. Namely, when trying to transform them to a defectless structure a high energy barrier occurs due to discontinuities, which inevitably take place at such a transformation. 12.3.5.1

Point Defects in a Two-Dimensional Nematic

We begin with a not particularly realistic example, a two-dimensional nematic. Here the classi®cation of defects is quite illustrative, and in the oneconstant approximation (12.20) a calculation of structures with point defects is simple. The equilibrium condition for a point defect, located at the center of the coordinate system, reads

388

G.P. Crawford, D. SvensÏek, and S. ZÏumer

` 2 y ˆ 0;

…12:27†

where y is the polar angle of the director ~ n ˆ …cos y; sin y†:

…12:28†

The solution to (12.27), depending only on the polar angle f, satisfying the continuity condition for the director, follows immediately as y n ˆ 0; G12; G1; G32; . . . : …12:29† y ˆ nf ˆ n arctg ; x The half-integral number n is the strength of the defect exactly as de®ned above. Note that in a medium with an ordinary vector order parameter ~ v with ~ v 0 ÿ~ v, e.g., a ferromagnet, the half-valued winding numbers are not allowed. The elastic energy of such a con®guration is obtained by integration of (12.20) for the solution (12.29):  2  2 ! … … 2p K y qy qy R r dr df ‡ …12:30† ˆ pKn 2 ln ; Fd ˆ 2 a qx qy r 0 0 R is a typical size of the nematic con®guration, whereas r0 is a microscopic cut-o¨ needed to avoid nonphysical divergence. This means that at distances near r0 the director con®guration (12.29) cannot possibly be correct. In this region the deformation becomes large, so that the Frank elastic theory ceases to be valid, and changes in the scalar order parameter s or even the biaxiality b must be taken into account. In ®rst approximation a core of radius r0 is invented with the nematic in the isotropic (melted) state, having then  0; r < r0 , …12:31† sˆ s0 ˆ const; r > r0 . Of course, this is not the con®guration with minimal free energy. The core energy Fc due to melting is Fc ˆ pr02 D f ;

…12:32†

where D f is the di¨erence in free energy densities of isotropic and ordered phase. By minimizing the total energy Fd ‡ Fc the radius of the core is set to s Kn 2 ; …12:33† r0 ˆ 2D f so it increases linearly with the strength of the defect n. The size of the core is comparable to the nematic correlation length x (12.15). It is very small if compared to the wavelength of light, so one can conclude that optics cannot be used for the investigation of defect cores. However, a better description of the defect core is gained by a precise numerical calculation based on the tensor order parameter [38]. In the case of

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

389

an n ˆ 1 defect, it demonstrates a uniaxial center with s < 0 and ~ n pointing out of the plane, followed by a biaxial ring around it, where the so-called eigenvalue exchange takes place, i.e., the central con®guration is transformed into a uniaxial one, but this time with s > 0 and ~ n lying in the plane, which at great enough distance from the core can be described by a radial director ®eld and s constant. In the case where multiple defects with strengths ni are present, on account of (12.27) being linear, the equilibrium con®guration is obtained simply by summing the solutions (12.29) for a single defect X X y ÿ yi ni fi ˆ ni arctg : …12:34† yˆ x ÿ xi i i The elastic distortion energy of two defects is then [27, p. 529]: Fd ˆ Fd1 ‡ Fd2 ‡ 2pKn1 n2 ln

R ; r

…12:35†

where r is the distance between the centers of the defects. The ®rst two terms stand for the energies (12.30) of single defects, whereas the third term represents the interaction energy. Evidently, defects with equally signed strengths repel each other, while those with opposite strengths are attracted. In this sense, the strength of the defect resembles the electric charge, indeed, in the one-constant approximation the analogy is almost perfect. Neglecting the dependence of the core radius r0 on the winding number n it is possible to write (12.35) in a slightly di¨erent way [27, p. 529]: Fd ˆ pK…n1 ‡ n2 † 2 ln

R r ÿ 2pKn1 n2 ln : r0 r0

…12:36†

Let us now discuss the basic properties of point defects in the twodimensional nematic. Our main interest will be in how topologically di¨erent defects are characterized and then how they combine. Point defects (or line defects in a three-dimensional nematic) have already been characterized by their winding number or strength n. For the two-dimensional nematic it turns out that defects of di¨erent strengths are topologically di¨erent [39], i.e., they cannot be continuously transformed into each other. What is more, two defects with strengths n1 and n2 can combine to form a defect with strength n1 ‡ n2 , i.e., in combining, the winding numbers are simply summed. Particularly it follows that two defects with opposite winding numbers can combine to form a defectless structure with zero winding number n ˆ 0. Even if the defects remain unannihilated, (12.36) shows that the logarithmic divergence is eliminated if n1 ‡ n2 ˆ 0, so the distortion energy of two opposite defects is small, provided, of course, that they are not very far apart. This is not di½cult to understand, because using a loop that encircles both defects a winding number n ˆ n1 ‡ n2 ˆ 0 is determined, which re¯ects a defectless structure with a low distortion energy outside the loop. Generally, defects with oppositely signed (not necessarily equal in magni-

390

G.P. Crawford, D. SvensÏek, and S. ZÏumer

tude) strengths will combine in order to reduce the distortion energy. On the other hand, it is energetically favorable for a defect with large strength to decay into defects with lower strengths, which then can move apart reducing the distortion energy. One has to be careful when applying (12.36) to this case, because it does not include the core energies and, particularly, disregards the fact that the size of the core varies with the winding number. Nevertheless, if the newly created defects are able to move apart su½ciently the total energy is decreased. A question of defect characterization arises if it is located at the bounding surface of the nematic sample (Figure 12.4(c)), because now the loop cannot surround the defect. In this case a convention is necessary to determine the winding number: the director ®eld is to be arti®cially continued beyond the surface (i.e., as a mirror image) and then the loop is continued as if there were no wall. The corresponding surface defect strength n 0 adequately describes the density of the director ®eld deformation, which is like that shown in Figure 12.4(a). Nevertheless, n 0 is twice as large as the bulk defect strength n corresponding to the situation where the defect is displaced in®nitesimally from the wall (Figure 12.4(b)) so that the encircling loop lies in the medium completely, n 0 ˆ 2n. 12.3.5.2

Line Defects in a Three-Dimensional Nematic

As mentioned, the solution and distortion energies obtained in two dimensions can also be applied to straight-line defects of in®nite length in the three-dimensional nematic, provided that the director ®eld is constrained to a planar con®guration (e.g., by an external ®eld in the case of negative susceptibility anisotropy (12.19)). In this case, the distortion energies ((12.30), (12.32), (12.35) and (12.36)) must be interpreted as the energies per unit length of the system. Regarding topological stability, however, linear defects in three dimensions behave quite di¨erently compared to the point defects in two dimensions, because continuous transformations changing the winding number by an integer are now possible [39]. This implies that topologically all defects with integer strengths are no defects at all, since they can be continuously transformed to a defectless structure. For the defect on Figure 12.3 such a transformation (a so-called escape in third dimension) is achieved by a p=2 out-of-plane rotation of directors when going from the boundary toward the center. Similarly, defects with half-integer strengths are continuously transformable into each other, thus being identical. Ultimately this means that in the three-dimensional nematic there exists only one topological line defect, we choose it to be the defect with winding number n ˆ 12 (Figure 12.4(b)) [39]. Again the combination law is simply the addition of winding numbers. With only one topological defect, though, there is little possibility left: two defects with strengths n ˆ 12 can combine to form a defectless structure with n ˆ 0.

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

(a)

(b)

391

(c)

Figure 12.4. Defects with (a) n ˆ 1 and (b) n ˆ 12 in the bulk. The director ®eld of the defect located at (c) the surface is the same as in (a) and is therefore assigned a surface strength n 0 ˆ 1. Topologically, however, it is equivalent to the defect in (b) with n ˆ 12 (courtesy of J. Bajc).

12.3.5.3

Line Defects in a Cholesteric

In cholesterics, defects are much more complicated due to the broken translational symmetry of the system. An undeformed helical structure in the cholesteric can be described by ~ n ˆ e^x cos…qz† ‡ e^y sin…qz†;

…12:37†

with the z-axis as the axis of rotation. In a deformed structure, (12.37) is valid only locally with the z-axis directed along the local twist axis ~ q. Thus, the local coordinate system is de®ned by three mutually perpendicular vectors ~ n, ~ q, and ~ q ~ n. Now three types of line defects are possible [30]. The so-called w defects are the same as in the two-dimensional nematic, only the directors are rotating as we move in the third dimension (along ~ q). With the w lines the ~ q ®eld is regular. With l and t defects (Figure 12.5), however, the ~ q ®eld is singular along the line of defect, while ~ n and ~ q ~ n are regular along the l and t lines, respectively. Because ~ n is regular along the t-line, its energy is lower than that of the other two lines. Topologically the situation is much more diverse now than it was in the nematic case. In cholesterics, four topological defects exist: defects of all three types …w; l; t† with odd strengths n ˆ 2k ‡ 1 (designated by C 0 ), plus defects with half-integer strengths n ˆ k ‡ 12 of types w, l, and t, respectively. Con®gurations of all three types with even strengths (labeled C0 ) are not topological defects. Combination rules for all these defects are collected in Table 12.1 [41]. For the ®rst time, nonuniqueness upon combination is encountered. According to Table 12.1, two l, t, or w defects of half-integer strengths can combine to form either a defectless structure or a defect with odd strength. The ®rst process can be visualized as a mergence of defects

392

G.P. Crawford, D. SvensÏek, and S. ZÏumer

(a)

(b)

(c)

(d)

Figure 12.5. Cross-sections of director ®elds corresponding to the l and t lines: l with (a) n ˆ 12 and (c) n ˆ ÿ 12; t with (b) n ˆ 12 and (d) n ˆ ÿ 12. Points indicate the director is perpendicular to the page, whereas marked ends represent tilted directors. The line defect is indicated by a heavy dot (courtesy of J. Bajc). Table 12.1. Results of combining two topological defects. Note that in some cases they are not unique, but depend on the path along which the defects are brought together. The l, t, and w defects carry half-integer strengths, while the C 0 defects are characterised by old strengths and can be of either kind …l; t; or w†. Topologically, the con®gurations l, t, and w with even strengths are defectless structures …C0 †

C0 C0 l t w

C0

C0

l

t

w

C0 C0 l t w

C0 C0 l t w

l l C0 or C 0 w t

t t w C0 or C 0 l

w w t l C0 or C 0

with n1 ˆ 12 and n2 ˆ ÿ 12 (topologically identical defects) to an n ˆ 0 structure. On the other hand, the second outcome can be seen as combining defects with n1 ˆ n2 ˆ 12 to a defect with n ˆ 1. In truth, in both cases just mentioned, either outcome is possible; starting with two n ˆ 12 defects, one of them can be easily transformed when approaching the other, and then anni-

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

393

hilated to n ˆ 0 (and vice versa with n ˆ 12 and n ˆ ÿ 12 defects). What will actually happen depends on the path over which the two defects are brought together.

12.4

Chiral Nematic Liquid Crystal Droplets

The ordering of chiral nematics in general con®ning geometries is a complex problem. Therefore we decide to choose the spherical droplet as an instructive but nontrivial example. To a great extent, this section is a summary of research covered by [34], [13], [14]. First, we give some general features of chiral structures con®ned in spherical cavities, enforcing a planar anchoring with no preferred direction (easy axis). As a result of such anchoring, a spherical structure is obtained with the helical axes (~ q's) pointing radially. It can be described by spherical chiral surfaces, de®ned as surfaces perpendicular to ~ q, thus joining points in the director ®eld with the same phase of rotation. Every chiral surface contains a two-dimensional director ®eld to be rotated due to chirality as the surfaces succeed. It is known that the director ®eld on a sphere cannot be defectless but inevitably features defects of total strength n ˆ 2 [40]. As a result of this a line defect of type w emerges, running from the center to the surface of the droplet. Figure 12.6 shows the two most known structures drawn on sequential spheres. The structure with a radial defect line of

Figure 12.6. Director ®eld in the radial (above) and the diametrical (below) structures, shown for three successive chiral surfacesÐspheres. The ®rst structure contains one n ˆ 2 w-line along the radius, whereas the second structure shows a vertical n ˆ 1 w-line along the diameter (courtesy of J. Bajc).

394

G.P. Crawford, D. SvensÏek, and S. ZÏumer

(a)

(b)

Figure 12.7. Two con®gurations of lowest energy on a nematic disk with strong tangential anchoring at the boundary: (a) bipolar with two n ˆ 12 defects and (b) monopolar with an n ˆ 1 defect. Because the defects appear at the surface they could have also been given double strengths n 0 , as discussed above (from [14]).

strength n ˆ 2, known also as the Frank±Pryce structure, is observed most often, while structures with a diametrical defect line of strength n ˆ 1 are observed less frequently (see [30], [13], [8], and references therein). The e¨ect of the applied electric ®eld on the droplet structure will be discussed; here only substances with negative dielectric anisotropies wea (12.19) will be of interest. In this case perpendicular alignment is favored (12.18), so that the helical axes tend to align with the ®eld (applying an electric ®eld to a cholesteric with positive wea would result in destabilizing the chiral order). In a strong ®eld limit helical axes become completely aligned with the ®eld, or, in other words, chiral surfaces are planes perpendicular to the ®eld. Now strong anchoring implies a defective structure with a total strength of n ˆ 1 on each chiral plane. Con®gurations of lowest energy are those with two n ˆ 12 defects (bipolar planar structure) or a single n ˆ 1 defect (monopolar planar structure), all of them pushed to the surface (Figure 12.7). According to our previous discussion of surface defects they could also have been given surface strengths n 0 ˆ 1 and n 0 ˆ 2, respectively. Again the w defect lines appear as Figure 12.8 suggests, this time spiraling on the surface, however. Consequently, their length is greater than that of the inner w lines in spherical structures, which results in the stability of the latter if there is no electric ®eld applied. For electric ®elds which are not extreme, however, intermediate structures must exist between the spherical and planar solutions. The transition induced by the ®eld has been observed to be continuous. In ®rst approximation the helical structure of the cholesteric is considered to be unaltered, so new chiral surfaces have to be found that would then give the intermediate structure in a weak electric ®eld. Flattened ellipsoids qualify as a natural generalization of the spheres (Figure 12.9(a)), but unfortunately such chiral surfaces are not equidistant, resulting in pitch variations. In ®rst approxi-

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

(a)

395

(b)

Figure 12.8. Strong electric ®eld limit: director ®eld on a sequence of chiral surfaces in the case of strong anchoring for (a) planar bipolar and (b) planar monopolar structures (courtesy of J. Bajc).

(a)

(b)

(c)

Figure 12.9. Modeling of intermediate chiral surfaces: (a) ¯attened rotational ellipsoids are not equidistant, so they are substituted by (b) equidistant disk-like surfaces, referred to as the oblate chiral surfaces. The length d is a measure of how much they are ¯attened, i.e., how much they deviate from spherical shape. In (c) their threedimensional layout is presented (from [14]).

mation, however, as mentioned above, this must not be allowed on account of the high energy costs. A disk-like surface with a rounded side (corresponding to the outer part of a toroid) is chosen as the proper chiral surface instead (Figure 12.9(b), (c)), yielding a structure with constant pitch. It shall be referred to as the oblate chiral surface. The length d is a measure of the extent to which the oblate surfaces di¨er from the spherical ones. The latter are obtained from the former by putting d ˆ 0, while the planar surfaces result when d > R. The transition from the structures in zero ®eld to structures in nonzero ®eld can thus be described by the parameter d…E†. Comparing the new intermediate chiral surfaces to the spherical ones, there are two important distinctions. First, spheres become degenerate to a point, whereas the oblate surfaces become ¯attened to a circle carrying a

396

G.P. Crawford, D. SvensÏek, and S. ZÏumer

planar director structure. Second, from a topological point of view, two different types of oblate chiral surfaces exist; for d ‡ h < R they are closed, thus topologically identical to spheres, whereas for d ‡ h > R they are cut by the bounding surface to what are topological circles. From the way the chiral surfaces have been set up it is clear that the rim of the central circular chiral surface corresponds to a l-line defect of strength 12 (discontinuous ®eld of helical axes), which e¨ectively results in strong tangential anchoring on the rim. Hence, the bipolar and the monopolar structures (Figure 12.7) are stable on the degenerate central chiral surface, just as in the case of planar chiral surfaces (strong electric ®eld) with in®nite anchoring (Figure 12.8), only that now the e¨ective ``in®nite anchoring'' is a topological consequence of how the oblate chiral surfaces have been set up. Recently, two models have been constructed [13], [14] trying to describe the intermediate con®guration in a weak electric ®eld by means of oblate chiral surfaces, each in its own approximation. An intuitive topological model that was ®rst developed does not go into determining the director con®guration on each of the chiral surfaces [13]. Instead it assumes in®nite anchoring on the droplet boundary and predicts the existence of inner and surface w-line defects using purely topological arguments. Within an improved model, the director ®eld is constructed directly from the known con®guration on the central chiral surface, while for simplicity only weak anchoring is considered [14].

12.4.1

Topological ModelÐStrong Anchoring Limit

The director ®eld on closed oblate surfaces (obtained for d ‡ h < R) must still contain defects with total strength n ˆ 2, just like the director ®eld on a sphere (Figure 12.6). Again this results in inner w lines with a total strength of n ˆ 2, starting from defects on the central chiral surface. From an n 0 ˆ 2n planar defect, w lines of total strength 2n must necessarily emerge. Sooner or later these lines arrive at chiral surfaces that are not closed, being topologically equivalent to disks (Figure 12.9). On account of the in®nitely strong anchoring assumed in this model, the director ®eld on these surfaces must contain defects of total strength n ˆ 1. To reduce the free energy they are pushed to the droplet surface, which results in w defect lines, spiraling on the surface. In this way, w lines are obtained both in the droplet and on its surface, as shown in Figures 12.10 and 12.11, thus both the features of spherical structures in the zero ®eld as well as those of planar structures in the strong ®eld are encountered in a single intermediate oblate structure. If the dielectric energy contribution is calculated for both spherical structures in a weak electric ®eld it is found that it is minimal when: in the diametrical spherical structure the diametrical w-line of strength n ˆ 1 is oriented along the ®eld; and in the radial spherical structure the radial w-line of strength n ˆ 2 is perpendicular to the ®eld.

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

(a)

397

(b)

Figure 12.10. Line defects in the (a) NDO and (b) RO structures. A low chirality …qR ˆ 4p† has been chosen in order to preserve clarity, d ˆ R=2. The l-line defect is shown dotted, whereas the inner and the surface w-line defects are represented by dashed and solid lines, respectively (courtesy of J. Bajc).

(a)

(b)

Figure 12.11. Line defects in the (a) PDO II and (b) PDO I structures. Again qR ˆ 4p, d ˆ 0:6R. The meaning of line textures is the same as in Figure 12.10 (courtesy of J. Bajc).

Two possible w-line orientations in the spherical structures (normal and parallel to the external electric ®eld) and two possible director con®gurations on the degenerate oblate chiral surface suggest four possible oblate structures: (1) The central circle carries the monopolar con®guration with an n 0 ˆ 2 defect, continued by a w-line with n ˆ 2 perpendicular to the ®eld as shown in Figure 12.10(b). This structure is known as the RO structure; clearly it originates from the radial spherical structure. (2) Two n ˆ 1 defect lines emerge from the n 0 ˆ 2 defect of the monopolar con®guration, both in the ``upward'' and ``downward'' direction (paral-

398

G.P. Crawford, D. SvensÏek, and S. ZÏumer

lel to the ®eld) as shown in Figure 12.11(b). This structure is formed from the diametrical spherical structure, its diametrical n ˆ 1 line moves to the side when the size of the central circle increases. It is known as the PDO I structure. (3) Two n 0 ˆ 1 defects of the bipolar planar structure are continued by two diametrical n ˆ 1 lines perpendicular to the ®eld (Figure 12.10(a)). This structure is referred to as the NDO structure, being a result of deforming the diametrical spherical structure, with the w-line perpendicular to the ®eld (a metastable orientation). (4) Two parallel n ˆ 12 lines emerge out of each n 0 ˆ 1 defect of the bipolar structure, oriented along the ®eld (Figure 12.11(a)). The so-called PDO II structure is evolved from the diametrical spherical structure by splitting its n ˆ 1 line into two n ˆ 12 lines. It is beyond our scope to pursue the total energy estimation of the oblate structures in a precise manner. Nevertheless, it has to be noted that only two energy contributions are taken into account in this model. These are the dielectric free energy (obtained by integration of (12.18)) and the elastic distortion energy or, to be more precise, merely an estimate of it, based on the distortion energy of straight defect lines in an in®nite medium (12.30), since the exact director ®eld con®guration is unknown. For a surface line defect the right strength entering (12.30) is that of n 0 , but approximately one-half of the director ®eld is cut away by the surface, yielding only one-half of the distortion energy. By means of total energy minimization the dependence d…E† is calculated, i.e., the measure of the extent to which the oblate structures are ¯attened on applying the electric ®eld (Figure 12.12). The only oblate structure that has been observed experimentally is the RO structure [43]. Comparison of the experimental values for d=R with predictions of the topological model shows a qualitative agreement, whereas quantitatively the correspondence is poor (Figure 12.13). The main disadvantage of the topological model is the assumption of in®nitely strong anchoring. Namely, on account of the strong anchoring, the surface w-line defects necessarily exist regardless of large energy contributions due to their considerable length. Indeed, with decreasing anchoring strength the surface w lines gradually disappear because a deviation from the preferred surface alignment costs less energy than the lines themselves. In a chiral system, ordering induced by the surface is particularly costly if it tends to disrupt the spontaneous twist deformation. For planar anchoring this happens as soon as the chiral axis ceases to be normal to the surface. Evidently this is the case in the oblate structures. The strong anchoring is a valid approximation only when l f lc is satis®ed, i.e., when the extrapolation length l (12.24) is small compared to the pitch length lc of the chiral helix (12.17), of course, l f R must still hold (see Figure 12.14). If l A 50 nm, which corresponds to a very strong anchoring, the approximation with in®nite anchoring is questionable for pitch lengths less than 0.5 mm.

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

399

Figure 12.12. Dimensionless measure of ¯atness of oblate structures (d=R) as a function of dimensionless electric ®eld strength E=E0 , E0 is chosen to be the ®eld at which the electric coherence length xe equals R. The dependencies for the four oblate structures are shown: RO (heavy solid line), NDO (solid), PDO I (dashed), and PDO II (dotted). Two sets of curves are shown, the upper one corresponds to qR ˆ 10p, the lower one to qr ˆ 40p (courtesy of J. Bajc).

Figure 12.13. Comparison of theoretically predicted transition from spherical to planar structure via the intermediate RO structure with experimental observations [43]. Dimensionless quantities Q ˆ qR are attached to the calculated curves, whereas the experimental data corresponds to Q A 20p. Two sets of experimental values are shown, obtained by observations of droplet textures parallel (squares) and perpendicular (circles) to the ®eld (courtesy of J. Bajc).

400

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.14. Director ®eld on successive planar chiral surfaces in the case of in®nite (left) and very weak planar anchoring. With the left con®guration the condition R A lc g le is ful®lled, whereas R g lc A le is valid for the right case. As the chirality is increased, the con®guration on a planar chiral surface approaches the undistorted one (right), and the surface w lines disappear (from [14]).

12.4.2

Director ModelÐWeak Anchoring Limit

As promised above in this model the director ®eld con®guration on oblate chiral surfaces will be constructed, whereas the anchoring on the droplet surface is considered to be very weak. According to the estimate made at the end of Section 12.4.1 the weak anchoring may give better results than the in®nite one assumed in the topological model (Figure 12.14). The director con®guration on the central circle of radius d (Figure 12.9) is the same as before, with the l n ˆ 12 line defect of strength on the edge e¨ectively implying in®nitely strong anchoring. In one-constant approximation (12.20) the two most probable planar con®gurationsÐthe bipolar and the monopolar structure (Figure 12.7)Ðcan be easily obtained in an analytical form. The bipolar director ®eld expressed in a cylindrical coordinate system reads ~ nbi ˆ

…d 2 ÿ r 2 † sin f …d 2 ‡ r 2 † cos f ^ ^ f; r ‡ d 4 ‡ r 4 ‡ 2d 2 r 2 cos…2f† d 4 ‡ r 4 ‡ 2d 2 r 2 cos…2f†

…12:38†

with the defects lying in the y-axis at f ˆ p=2 and f ˆ 3p=2, respectively (see Figure 12.15(a)). The monopolar solution is obtained simply by considering the director ®eld solution around a defect of strength n ˆ 2 in an in®nite medium (12.29). In the one-constant approximation it consists of circles of di¨erent radii having one point in common. If this point is put on the edge of the central circle and the structure is properly oriented so that the circle of radius d matches the rim of the central circle, both the distortion energy is minimized and the boundary condition is obeyed (Figure 12.7(b)), hence this must be the solution for the monopolar structure …r 2 ÿ d 2 † cos…f† …r 2 ‡ d 2 † sin f ÿ 2dr ^ ~ nmo ˆ q r^ ‡ q f: d 2 ‡ r 2 ÿ 2d sin f d 2 ‡ r 2 ÿ 2dr sin f The defect of strength n 0 ˆ 2n ˆ 2 lies on the y-axis at f ˆ p=2.

…12:39†

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

(a)

401

(b)

Figure 12.15. (a) A cylindrical coordinate system with coordinates …r; f; z† is used to describe the director ®eld on the central part of the chiral surfaces, while for the outer part a toroidal coordinate system with coordinates …r; y; f† is particularly suitable (from [14]).

Now the director ®eld on all chiral surfaces can be constructed from the solutions (12.38) and (12.39) on the central circle. The chirality is to be kept ®xed, so the surfaces in Figure 12.9(b) and (c) serve as suitable chiral surfaces. As mentioned, the anchoring on the droplet boundary is said to be weak, so that in the ®rst approximation the director ®eld, anywhere in the droplet, is determined solely by the central circle con®guration. Most conveniently, the director ®eld on the central (planar) part of the chiral surfaces is expressed in a cylindrical coordinate system (Figure 12.15(a)), whereas for the outer (bent) part a toroidal coordinate system is used (Figure 12.15(b)). By de®nition, chiral axes are perpendicular to chiral surfaces, so their con®guration is like the one presented in Figure 12.16(a). Now only the constant chirality requirement has to be satis®ed. Directors in points B and B 0 (Figure 12.16(b)) are uniquely determined by the directors in A and A 0 , respectively. The director in B, for example, is rotated with respect to the director in A by an angle j ˆ q AB, depending on the chirality q and the distance between the points; and analogously for points B 0 and A 0 . In the central part the director ®eld is expressed as ~ nˆ

sin…f ÿ qz† ÿ %p2 sin…f ‡ qz† cos…f ÿ qz† ‡ %p2 cos…f ‡ qz† ^ …12:40† q r^ ‡ q f; 1 ‡ %p4 ‡ 2%p2 cos…2f† 1 ‡ %p4 ‡ 2%p2 cos…2f†

using the cylindrical system with %p ˆ r=d being a dimensionless radial coordinate. For the set of points fr ˆ d; z; f ˆ Gp=2g the director ®eld given by (12.40) is singular, resulting in two inner w lines of strength n ˆ 12 parallel to the z-axis. Evidently, the con®guration so obtained appears like the PDO II structure. In the outer part, the director ®eld expressed in the toroidal system depends only on the distance from the edge of the central circle ^ ~ n ˆ ÿsin…qr†y^ ‡ cos…qr†f:

…12:41†

G.P. Crawford, D. SvensÏek, and S. ZÏumer

402

(a)

(b)

Figure 12.16. (a) Layout of chiral axes. The ®eld of chiral axes is homogeneous in the central part, whereas in the outer part it is splayed. (b) The directors in points B and B 0 are simply rotated directors of A and A 0 ; chirality is kept constant (courtesy of J. Bajc).

The PDO I structure is constructed starting with the monopolar planar con®guration by the same procedure as the PDO II structure. In the central part the solution is ~ nˆ

%p2 cos…f ‡ qz† ÿ cos…f ÿ qz† ‡ 2%p sin…qz† 1 ‡ %p2 ÿ 2%p sin f ‡

r^

%p2 sin…f ‡ qz† ‡ sin…f ÿ qz† ÿ 2%p cos…qz† ^ f; 1 ‡ %p2 ÿ 2%p sin f

…12:42†

whereas in the outer part the solution is ^ ~ n ˆ sin…qr†y^ ÿ cos…qr†f:

…12:43†

Now a singularity is obtained for fr ˆ d; z; f ˆ p=2g, representing the inner w-line with n ˆ 1. In what follows solutions for the other two structures (the RO and NDO) with inner w lines perpendicular to the ®eld must be sought. The NDO structure can be constructed from the PDO II structure by a continuous transformation, because both have the same structure on the central circle. All that has to be done is to combine both pairs of w lines of strength n ˆ 12 oriented parallel to the ®eld to two single w lines with n ˆ 1 lying in the central plane, as shown in (Figure 12.17). Topologically such transformation is allowed (Table 12.1, page 392). Every chiral plane can be regarded as an elastic membrane, since the director ®eld on it is governed by elastic forces. Now the desired transformation can be viewed as stretching of the membrane, as Figure 12.17 suggests, so that the director ®eld of the central part only is drawn over the whole chiral plane of the NDO structure so obtained.

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

(a)

403

(b)

Figure 12.17. Continuous transformation of (a) the PDO II structure to (b) the NDO structure. Above a three-dimensional director ®eld sketch is shown for a speci®c chiral surface, whereas below transformation of defect lines is presented schematically; their strength is represented by line thickness (from [14]).

For details see [14]. The director ®eld on the central part of the NDO structure is ~ nˆ

sin…f ÿ qz† ÿ %n2 sin…f ‡ qz† cos…f ÿ qz† ‡ %n2 cos…f ‡ qz† ^ f; …12:44† p r^ ‡ q 1 ‡ %n4 ‡ 2%n2 cos…2f† 1 ‡ % 4 ‡ 2% 2 cos…2f† 2

n

where %n ˆ h…z†r=d ˆ h…z†%p and h…z† ˆ d=…d ‡ pjzj=2† is a shrinking factor. The director ®eld on the outer part of chiral surfaces is expressed as sin…f ÿ qr† ÿ h 2 …r; y† sin…f ‡ qr† ~ n ˆ p y^ 1 ‡ h 4 …r; y† ‡ 2h 2 …r; y† cos…2f† cos…f ÿ qr† ‡ h 2 …r; y† cos…f ‡ qr† ^ ‡ p f; 1 ‡ h 4 …r; y† ‡ 2h 2 …r; y† cos…2f†

…12:45†

with the same shrinking factor h…r; y† ˆ …d ‡ ry†=…d ‡ rp=2†, written in toroidal coordinates this time. The director ®eld is singular in fr > d, y ˆ p=2; f ˆ Gp=2g, giving the w-lines of strength n ˆ 1 that lie in the equatorial plane. Analogously the RO structure is created, with the solution for the central part ~ nˆ

%n2 cos…f ‡ qz† ÿ cos…f ÿ qz† ‡ 2%n sin…qz† r^ p 1 ‡ %n2 ÿ 2%n sin f ‡

%n2 sin…f ‡ qz† ‡ sin…f ÿ qz† ÿ 2%n cos…qz† ^ p f; 1 ‡ %n2 ÿ 2%n sin f

…12:46†

404

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.18. Director ®eld on three successive chiral planes of the RO structure, d ˆ R=2, qR ˆ 2p, whereas values of h are h ˆ 0, h ˆ d=4, and h ˆ d=2, passing from left to right (from from [14]).

and the one for the outer part ~ nˆ

h 2 …r; y† cos…f ‡ qz† ÿ cos…f ÿ qz† ‡ 2h…r; y† sin…qz† ^ y p 1 ‡ h 2 …r; y† ÿ 2h…r; y† sin f ‡

h 2 …r; y† sin…f ‡ qz† ‡ sin…f ÿ qz† ÿ 2h…r; y† cos…qz† ^ f: p 1 ‡ h 2 …r; y† ÿ 2h…r; y† sin f

…12:47†

The director ®eld on three successive chiral planes of the RO structure is shown in Figure 12.18. With the help of Figure 12.19 we summarize schematically how continuous transformations between the four oblate structures can be performed. As in the case of the topological model the stability of the four calculated con®gurations is assessed by their total energies, only that with the director ®elds known the energies can be calculated in a much more precise way. The energy contributions to include are the elastic distortion energy, the core energy of defect lines, the dielectric energy, and the surface energy due to deviations from the favored anchoring direction. The minimum of the total energy with respect to d=R for di¨erent values of the electric ®eld E yields a function d=R…E†, shown in Figure 12.20 for all four structures in a high chirality limit. All curves are quite alike, the di¨erences appear to be relevant only at low electric ®elds, i.e., at early stages of the transition. As observed, the transition via the PDO I structure starts at a remarkably higher ®eld strength than the others. The PDO I structure originates from the diametrical spherical structure (like the PDO II), then being transformed to what resembles the RO structure. The latter, however, starts directly from the radial spherical structure. For this reason the PDO I structure undergoes stronger elastic deformation at the beginning of the transition, so higher ®elds are needed in order to induce the transformation. In the low chirality cases, a step-like behavior is observed (Figure 12.21), although globally it has little signi®cance.

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

405

Figure 12.19. Schematic presentation of continuous transformation between the oblate structures. The strength of line defects is shown by line thickness (courtesy of J. Bajc).

On the other hand, the strength of surface anchoring is more important. As expected, with stronger anchoring, larger ®elds are needed to induce the transition (Figure 12.22). The free energy estimate [14] gives the lowest value for the PDO II structure and the highest one for the RO structure. This is consistent with earlier estimates [8] yielding lower free energy for the spherical chiral structure with diametral defect line in comparison to the one with radial …n ˆ 2† defect line. It also agrees with a simple topological consideration, according to which an n ˆ 2 line has higher energy than four n ˆ 12 lines. However, the fact that solely the RO intermediate structure has been observed experimentally comes as a surprise. A partial source of error is the simple modeling of line defects, i.e., by a core of isotropic phase (12.31); their free energy is too high, and, in particular, the interaction between the w lines is not properly described for small separations. Further, possible reduction of the free energy not taken into account is the fact that for integral strengths the core can disappear to form an escaped structure [7], [8].

406

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.20. Stability diagram, showing the electric ®eld-induced transition from spherical to planar structures via the four intermediate oblate structures in the case of a high chirality qR ˆ 60p. For all structures the behavior is similar, except for the low ®eld region at the beginning of the transition (from [14]).

Figure 12.21. In¯uence of chirality on the form of d=R…E† for the RO (on the left) and the PDO II structure. The chiralities are: qR ˆ 10p (heavy), qR ˆ 20p (thin), and qR ˆ 60p (dashed) (courtesy of J. Bajc).

Although the model reported cannot provide relevant free energy values it can well explain the observed transformation of the spherical chiral structure to a nearly planar one. Experimental data for d=R…E† can be best ®tted with an anchoring strength W0 of 0.2±0.4 mJ/m 2 (Figure 12.22). An even more direct assessment of theoretical results has been established by simulating the polarizing microscope textures for the RO structure and comparing them to

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

407

Figure 12.22. In¯uence of surface anchoring strength on the transition via the RO structure and comparison with experimental data [43], observed parallel (squares) and perpendicular (circles) to the ®eld. The theoretical curves have been calculated taking Kii ˆ K ˆ 5
10ÿ12 N, wea ˆ ÿ5, qR ˆ 20p, and R ˆ 10 mm. The anchoring strength W0 (measured in mJ=m 2 ) is given with the curves. If compared to the experiment the right value of W0 should be between 0.2 and 0.4 mJ/m 2 (from [14]).

(a)

(b)

Figure 12.23. Comparison of simulated (on the left) and observed polarizing microscope patterns for the RO structure, viewed from the direction (a) normal to both the electric ®eld and the w-line and (b) along the ®eld. On the photograph in (a) one of the chiral planes is emphasized in order to perceive the others. The radial w-line is visible in (b), whereas in (a) it causes a left±right asymmetry (from [14]).

textures observed experimentally [43]. The textures have been calculated considering only the rotation of polarization due to the anisotropy wea , whereas di¨raction as well as refraction e¨ects have been neglected. Both calculated and observed microscope images are shown in Figure 12.23, demonstrating a nearly perfect agreement. Of course, one cannot expect an exact matching of bright and dark patterns, because they depend on the composition of light. In the simulated images only three di¨erent wavelengths have been used to simulate ordinary white light worked with in experiments. For more details, see [14]. In Section 12.6, dedicated to applications, there can be found more information on the electrically controlled re¯ectivity from the PDLC made of chiral droplets.

408

G.P. Crawford, D. SvensÏek, and S. ZÏumer

12.5

Polymer Networks Dispersed in Liquid Crystals

Polymer networks which can memorize the orientational order of the nematic liquid crystal environment where they are assembled [71], [72], [73], [74] are particularly attractive because of their potential for a variety of electrooptic technologies. We postpone this subject to the last section and here concentrate our attention on the ordering and structures of these composite materials. These systems have many physical properties analogous to liquid crystals con®ned to di¨erent submicrometer-sized cavities [75], [76] and random porous matrices [77], [78]. Large surface-to-volume ratios enable a strong in¯uence of the polymer network on nematic ordering in the liquid crystalline solvent and thus govern optical properties of the composites. The information about these systems obtained from diamagnetic and viscosity measurements [79], birefringence [72], [74], small angle neutron scattering measurements [73], nuclear magnetic resonance and relaxation [80], and scanning electron microscopy [81], patched together, led to a consistent picture about the polymer structure on all scales. In the following we focus our attention to the most complete study of birefringence so far [74] instead of completely reviewing all research on polymer dispersions in liquid crystals. We show how one can, with a rather simple experimental method combined with the phenomenological description of liquid crystalline ordering, point out important details about the polymer network structure on both the micro and macro levels.

12.5.1

Measurements of Optical Anisotropy

The monomer BMBB-6 (4 0 4-bis-f4-[6-(methacryloyloxy)-hexyloxy] 0 benzoateg-1,1 biphenylene) and very small amounts of photoinitiator BME were mixed with 5CB to perform a study of the polymer network-induced birefringence in the isotropic phase of a nematic liquid crystal. Mixtures of 5CB in the nematic phase with 1, 2, 2.5, 3, and 4 wt.% of monomer were ®lled into planar cells and irradiated by UV light (4 W/cm 2 ) at constant ambient temperature for 1 hour. In some cells liquid crystal was substituted by hexane which, after the cells were opened, completely evaporated so that only the polymer network was left on the glass substrate. The examination by SEM (Figure 12.24) shows a ®ber-like polymer network perpendicular to the substrate. The thickness of ®bers and aggregates of ®bers ranges from 0.1 mm to 1 mm. Although the substitution of the liquid crystal with the solvent hexane [72] does not a¨ect the network this is certainly not true for the evaporation of the solvent. The rather dense SEM-detected structures are the result of shrinking and partial collapse of the network after the solvent was removed. The structure close to the surface which was less e¨ected by the drying process indicates that ®bers of thickness @ 0:1 mm after evaporation combine in thicker tree-trunk-like structures.

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

409

Figure 12.24. Polymer network originally formed in a homeotropically oriented liquid crystal and examined with an electron microscope after the removal of the liquid crystal (from [74]).

The cells of all ®ve polymer concentrations were used for the birefringence study with the light of an He±Ne laser. The polarizer and analyzer were crossed, and the cell was aligned so that the rubbing direction (director ®eld ~ n) was tilted by 45 with respect to the polarizer and analyzer. The intensity of light with wavelength l transmitted through a cell of thickness d is described by I ˆ I0 sin 2 …pDnd=l†:

…12:48†

This expression allowed us to determine the temperature-dependence of the e¨ective birefringence Dn within a precision of 10 mK. Experimental results can be summarized as follows [74]. Isotropic Network in a Liquid Crystal Phase. The photopolymerization in the isotropic phase yields in a polymer network exhibiting high light scattering when cooled to the nematic phase but it was completely transparent and did not exhibit birefringence above the NI transition temperature. This experiment reveals that the polymer network assembled in the isotropic phase does not possess any long-range order. If there is a local order, its range is small compared to the wavelength of light. Therefore its e¨ect is completely averaged out by a light beam sampling randomly oriented areas of local order.

410

G.P. Crawford, D. SvensÏek, and S. ZÏumer Figure 12.25. Experimentally determined temperature-dependence of birefringence of (a) a polymer network in an isotropic liquid crystal and (b) a polymer network in an isotropic solvent for several polymer concentrations (from [74]).

Ordered Network in an Isotropic Liquid Crystal. The strong pretransitional increase of the e¨ective birefringence (Figure 12.25(a)) for all examined concentrations of the polymer suggests that in addition to the direct contribution of the polymer network, there is a temperature-dependent contribution from the paranematic order induced in the isotropic liquid crystal phase by internal surfaces of the network. Ordered Network in an Isotropic Solvent. Substituting the liquid crystal which surrounds the network with the isotropic ¯uid chlorobenzene and assuming that the birefringence DnPIL can solely be attributed to the polymer network, it was estimated to be between 5
10ÿ4 to 3
10ÿ3 depending on the network concentration h (See Figure 12.25(b)). The weak temperaturedependence indicates that networks are practically rigid and stable up to

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

411

Figure 12.26. Schematic presentation of the polymer network in a two-scale (®bril-bundle) model showing both the local and macrodirector. The local distribution of ®brils is represented by a square array of polymer ®brils. The relevant distances are also illustrated (from [74]).

100  C. Not detecting any appreciable change in the volume of the dispersion when the liquid crystal was replaced with the chlorobenzene, one can assume that the e¨ective order parameter of the network does not change as well.

12.5.2

Model Structure of the Network

The polymer network dispersed in a liquid crystal is modeled by an array of thin polymer ®brils formed along the local director ®eld. The ®brils are described locally as parallel cylindrical rods characterized by radius R and packed into a two-dimensional square array with inter-®brile distance d (Figure 12.26(a)). The local order parameter sp of the polymer network is also assumed to be equal to the order parameter sn of the bulk nematic liquid crystal where it was formed. Further, assumed to be the polymer-induced paranematic order is uniaxial with the director ~ nloc parallel to the polymer ®brils. This allows us to describe the paranematic ordering in the inter®brile space by a simple scalar order parameter ®eld s (see on (12.3)). To take into account the observations [74], [73], [79], [81], indicating the existence of ®berlike objects with diameters around 0.1 mm, the thin ®brils with typical diameter D are assumed to form bundles of polymer-rich material where the concentration h 0 ˆ pR 2 =d 2 is larger than the average polymer concentration h. Further, a bundle of parallel ®brils forming a large (micron) scale network is simply represented by an average interbundle distance B (Figure 12.26(b)). In the space between bundles the polymer concentration is low and thus does not contribute to the paranematic ordering. It should be stressed that in the nematic phase the behavior of these parts of liquid crystal is characterized by

412

G.P. Crawford, D. SvensÏek, and S. ZÏumer

constraints on the scale of the interbundle distance B while the behavior of the liquid crystal in the bundles is characterized by the much smaller inter®brile distance d. Therefore only liquid crystal in the polymer-poor regions can be easily a¨ected by an applied electric ®eld and is thus useful for electro-optic applications. This ®bril-bundle picture is denoted as a two-scale model of the polymer network. To reduce the freedom (four parameters) the bundle thickness D will be taken as 60 nm using the results of Jakli et al. [73], [79] which is also in agreement with the SEM study [81]. Further, will be used the ratio h 0 =h of the polymer concentration in the bundle (h 0 ) to the average concentration (h) as a parameter instead of the interbundle distance B. Thus the two-scale model is controlled by only two independent structural parameters R and h 0 =h.

12.5.3

Optical Anisotropy

When a light beam passes through a sample with nonuniform direction and degree of ordering it is refracted and scattered. The birefringence of a uniaxial medium is proportional to the orientational order parameter Dn ˆ s Dn0 if n g Dn0 (see, for instance, [25]). The refraction which is linear in the order parameter is expected to be crucial in the isotropic phase, while scattering e¨ects which are quadratic in order parameter variations should be more carefully examined. To estimate the relevance of scattering on small domains with size D f l one can use the Rayleigh or Rayleigh±Gans description [83] for the scattering cross section of such a domain yielding s @ …Dn0 s† 2 D 6 =l 4 . Taking Dÿ3 for the density of randomly oriented domains and following ZÏumer et al. [82] one can, for small domains D U 0:1l, ®nd that the extinction coe½cient is negligible even in a nematic liquid crystal phase where Dn0 s @ 0:2 yields s=D 3 @ <10ÿ4 mmÿ1 . The scattering of the domains with D @ l can be treated using anomalous di¨raction approximation [83]. One can estimate the cross section of such a domain as s @ …Dn0 s† 2 D 4 =l 2 and the corresponding extinction coe½cient: s=D 3 @ 1 mmÿ1 in the nematic phase and 0.01 mmÿ1 in the isotropic phase. This explains strong light scattering in the nematic phase and practically no scattering in the isotropic phase. The e¨ective birefringence in the partially ordered isotropic phase is thus simply given by an average Dn ˆ

1 V

… V

Dn0 s…~ r † d 3r

…12:49†

where V is the volume sampled by the beam used in probing birefringence. On macroscale the polymer network is disordered in addition to the local scale disorder of the polymer forming ®brils. Therefore it is useful to sepa-

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

413

rate the global average h i in two independent averages: h iloc over local variables and h imacro over macrovariables. The global order parameter of the network alone hspol i, can be thus written as hspol i ˆ smacro sn ;

…12:50†

where smacro describes macroscale order. Similary the global liquid crystal order parameter hsLC i can be written as hsLC i ˆ smacro s;

…12:51†

with s standing for the local (within a bundle) average of the liquid crystal order parameter. In the next section s will be calculated for the geometrical model described above, assuming that polymer surfaces induce paranematic ordering characterized by the surface preferred value sn and an unknown coupling strength G: Assuming that densities and optical properties of both materials do not di¨er much, and using relations (12.50) and (12.51) one can write Dn ˆ Dn0 smacro ‰…1 ÿ h†s…G; sn † ‡ hsn Š;

…12:52†

where Dn0 stands for the e¨ective index of refraction of the perfectly ordered material and h for the percentage of polymer in the mixture.

12.5.4

Paranematic Ordering Induced by the Network

In our simpli®ed picture the minimization of the Landau±de Gennes free energy composed of bulk, gradient, and surface terms (see Section 12.3) yields the following second-order di¨erential equation a…T ÿ T  †s ÿ bs 2 ‡ cs 3 ÿ 32 L` 2 s ˆ 0;

…12:53†

and the boundary condition at the surface of the polymer ®brils 3 r jˆR; ? 2 L…`s†j~

ÿ 2G…s ÿ sn †j~r jˆR ˆ 0:

…12:54†

The symbol ? stands for the direction perpendicular to the surface of the polymer ®brile. The elastic constant L has a value of L ˆ 1:1
10ÿ11 J/m. The constant G, describing the strength of the surface coupling, will be our third parameter in the calculation of the average order parameter s. In the case of a very diluted polymer network or at high temperatures liquid crystal is ordered only in the vicinity of the polymer ®bers. If the order parameter s is small the quadratic and cubic term in (12.53) can be neglected and an analytical solution can be expressed with the modi®ed Besselp func tions K0 …r† and K1 …r† [74] which asymptotically leads to s…r† z eÿr=x = r=x. For a particular case, pro®les are shown in Figure 12.27. In the case of a concentrated polymer network modeled by the square lattice (inset in Figure 12.26) a solution for the order parameter pro®le (see Figure 12.28) is found numerically [74]. It can be seen that for thin ®brils

414

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.27. Order parameter pro®le s…r† for di¨erent temperatures in the vicinity of the phase transition [74]. The parameters are R ˆ 2:5 nm, G ˆ 0:01 J/m 2 , and sn ˆ 0:3 (from [74]).

Figure 12.28. The order parameter pro®le in a diagonal direction of a square array of parallel ®brils with radius 1 nm and inter®bril distance 8.9 nm for several temperatures. The parameters of the model are the same as in Figure 12.27 (from [74]).

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

415

(R ˆ 1 mm) and low polymer concentrations, even well beyond the transition point, the parameter s between ®brils is too high for a single ®bril approximation.

12.5.5

Comparison of Calculated and Experimental Birefringence with Experimental Data

The order parameter of the polymer network hspol i is estimated from the experimentally determined birefringence DnPIL of a polymer network dispersed in the isotropic solvent (chlorobenzene) using (12.52) with Dn0 ˆ 0:3 and s ˆ 0. For the polymer network concentrations between 0.01 and 0.04, one ®nds that hspol i is between 0.15 and 0.28. These values are substantially smaller than sn A 0:5 of the liquid crystal at room temperature where the network was formed. The macroscale disorder smacro introduced with (12.50) is estimated to vary between 0.3 and 0.5 depending on the network concentration. The largest value achieved at 3% concentration indicates that networks at lower concentrations are probably too fragile to retain complete order of the initial environment where they were formed. On the other hand, with increasing concentration, more cross-linking is expected and this certainly leads to the decrease of the network orientational order. The BMBB6 network exhibits much higher order as compared to the BAB type [72]. Under SEM (Figure 12.24) the network looks like being built of more or less oriented ®bers, while in the BAB case the structure looks like being built of globules not showing any particular order [72]. For the macro order parameter smacro obtained above the paranematic temperature-dependence of the birefringence is calculated from (12.52) and compared to the experimental data. The average s needed in (12.52) is calculated for a chosen set of parameters d; h 0 =h. This was achieved by solving the Euler±Lagrange equation (12.53) for the order parameter pro®le s…x; y† in a square lattice array of ®brils characterized by the distance d (see inset in Figure 12.26), and taking into account the ratio h 0 =h of intrabundle and average polymer density. In Figure 12.29(a) and (b) theoretical birefringence is compared with the experimental data for network concentrations 1, 2, 2.5, and 3%. The calculated birefringence is in the best agreement with the experimental data if the radius of the polymer ®bers is around 2 nm (1.8 nm, 2 nm, 2.2 nm, and 2.4 nm for concentrations of 1%, 2%, 2.5%, and 3%, respectively) and the same surface coupling constant G is about 0.01 J/m 2 . For a 1% concentration of polymer ®bers, the experimental error is larger than for concentrations of 2% and 2.5%, because the induced birefringence decreases with decreasing concentration which results in a higher uncertainty of the data. Figure 12.29(b) clearly shows how sensitive the ®t is to the change of the ®bril radius. The values for a good ®t are approximately twice as large as the ones obtained for BAB [72]. Figure 12.29(c) shows the ®t for 4% polymer concentration performed with two di¨erent intra-bundle con-

416

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.29. The calculated birefringence (line) and experimental data (dots) for various temperatures and concentrations of the polymer. (a) For h ˆ h 0 ˆ 0:025, 0.02, and 0.01, the radius of the polymer ®brils corresponding to the best ®t is equal to 2:2 nm, 2 nm, and 1:8 nm, respectively. Parameters G ˆ 0:01 J/m 2 and sn ˆ 0:5 are equal for all concentrations. (b) For the polymer concentration 0.03, it is shown how sensitive the ®t is to the variation of the ®bril radius. (c) For the h ˆ 0:04 case, the comparison of the ®ts obtained for h 0 ˆ h and h 0 ˆ 2h is presented (from [74]).

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

417

centrations. A good ®t is obtained with R ˆ 1:5 nm and h 0 ˆ 2h. Therefore, with the initial choice for the bundle thickness, one estimates the interbundle distance to be approximately 0.1 mm. The relatively large polymer±liquid crystal coupling constant G [76] indicates that the induced liquid crystalline order parameter at the polymer surface is weakly temperature-dependent, which is consistent with the ®ndings of the NMR relaxation study [80]. Altogether, one can conclude that the presented visualization of the polymer network morphology as a network of bundles of thin ®brils describes reasonably well all experimental data. The basic building blocks of the networkÐthe thin polymer ®brils and their very ®ne distribution are the main cause of pretransitional birefringence [74]. On the other hand, the network of bundles, characterised by the submicrometer interbundle distance, in responsible for polymer stabilisation of nematic and chiral nematic structures and the corresponding electrooptical properties of the dispersions.

12.6

Display Applications

With the arrival of the information age, ¯at panel displays are playing a crucial role as the primary interface between microprocessor and human beings. Computer monitors, laptop screens, and television sets are a few examples. The trend of ¯at panel displays is toward thinner, lighter, and more energy-e½cient displays to satisfy the tremendous demand in the portable electronics industry such as notebook computers, palmtop personal computers, personal digital assistants, electronic books and papers, pagers, portable fax machines, portable information terminals, etc. Among all of the ¯at panel display technologies, ®eld emitter displays (FED), organic lightemitting diodes (OLEDs), plasma display panels (PDP), electroluminescence (EL), and liquid crystals displays (LCDs), LCD technology is expected to be the dominant one in the foreseeable future owing to its maturity and current industry infrastructure. Here we will discuss the use of cholesteric liquid crystals in the context of re¯ective display applications (see Figure 12.30), which is arguably the most promising application of these chiral nematic materials [19], [23]. The most common type of ¯at panel display is the transmissive LCD used in conjunction with a laptop computer where a certain amount of power is dedicated to the scan drivers, data drivers, controller, grayscale drivers, and a signi®cant portion (more than one-half ) is dedicated to the internal source of illumination (i.e., backlight). Since chiral nematic materials re¯ect light according (see, for instance, [25]) to the Bragg condition l ˆ hniP for normally incident light, where hni is the average index of refraction, and have a full-width at half maximum governed by Dl ˆ PDn, where Dn is the birefringence of the chiral material, they can e¨ectively re¯ect and modulate the ambient lighting, thereby removing the need for a power-consuming backlight. There are several di¨erent types of chiral nematic con®gurations for displays that

418

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.30. Comparison of transmissive and re¯ective displays.

will be discussed below, but before we engage in discussions about chiral displays, it is worth mentioning that some chiral nematic displays have bistable memory. Bistability is a much sought-after property of display materials that makes possible the use of an inexpensive passive addressing scheme for high-de®nition ¯at panel displays. Crooker and Yang [9] were the ®rst to make use of a polymer dispersed cholesteric liquid crystal (PDCLC). A schematic illustration of the PDCLC is depicted in Figure 12.31. In the zero-voltage state, the chiral nematic droplets are organized in a spherical structure with a radial defect line (see Section 12.4 for details) and therefore do not selectively re¯ect incoming ambient lightÐonly a slight light scattering is observable. Upon application of a voltage the helical axes rearrange themselves to be perpendicular to the substrate (planar state with planar chiral surfaces, see Section 12.4) if the dielectric anisotropy of the chiral material is negative. At this stage, selective re¯ection occurs according to the Bragg condition as in bulk. Upon removal of the applied voltage, the planar texture relaxes back to the radial helix con®guration (i.e., no bistable memory). The color of the selectively re¯ected light can be controlled by adjusting the concentration of the liquid crystal mixture. Kitzerow and Crooker [10], [24] adjusted the chiral concentrations to tune the peak wavelength of PDCLCs for the red, green, and blue primary colors of a full-color display. Kato and coworkers [45], [46], [47] achieved full-color by integrating red, green, and blue stacked devices. Because of the right- or left-handed helical nature of the planar state of a PDCLC, the re¯ection peak is theoretically limited to 50% at the Bragg wavelength (in practice, it is often much less). This is why Kato and coworkers stacked right- and left-handed twisted displays to increase the overall re¯ectivity of the PDCLC. These chiral PDLC materials, however, do have high drive voltages (@50 V) [9], and typically do not come near to the 50% re¯ectivity

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

419

Figure 12.31. Schematic illustration of a PDCLC.

achievable theoretical limit. They usually re¯ect in the 10±20% range and therefore their usefulness in displays is limited due to poor brightness and high drive voltages. In 1992, Yang and coworkers [18] reported another attractive feature of cholestertic nematic materials. Instead of polymer-dispersed materials that encapsulate the chiral liquid crystal, they disclosed a new con®guration using only small concentrations of polymer (@2 wt.%), that could e¨ectively scatter white incident light. The polymer forms an interpenetrating network structure throughout the bulk material (for details see Section 12.5). The other di¨erence is that the value of the pitch is selected such that the cholesteric liquid crystal material re¯ects light in the infrared region. This type of chiral technology is optically reminiscent of the conventional PDLC, except it can be tailored to operate in the reverse and normal mode, and is essentially haze-free [18], [48]. In the reverse mode display, the glass substrates are treated with a polyamide and rubbed, just like a conventional nematic LCD, and assembled in a uniform homogenous con®guration. The liquid crystal is doped with a low reactive monomer (@1±2 wt.%) and assumes a planar con®guration with helical axis normal to the substrates. The display is irradiated with UV light and the network polymerizes within the planar con®guration, and, after polymerization, the planar con®guration remains stable (see Section 12.5). The con®guration is depicted schematically in Figure 12.32. In the zerovoltage state, the cell appears totally transparent to visible light because the pitch is tuned in the IR. When a voltage is applied across the cell, the planar texture transforms into the focal conic texture (see, for instance, [53]), and the material is scattering (opaque) for all polarizations of the incident light. When the voltage is turned o¨, the focal conic texture relaxes back to the planar textureÐtherefore there is no bistable memory. The function of the

420

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.32. A schematic illustration of a reverse-mode polymer-stabilized cholesteric display. The display scatters white light in the voltage-on state (right) and is transparent in the o¨ state (left).

polymer in the reverse mode con®guration is to create focal conic domains that scatter light and force the chiral liquid crystal to return to the planar state after the voltage is turned o¨. The switching voltage of the reverse mode display is about 13 V where the liquid crystal begins to transform into the focal conic texture, to 20 V where the transmission decreases to 4% of its maximum transmission for a 10 mm cell gap. There is a signi®cant hysteresis associated with the reverse mode cell. The dynamic response times are impressive for the reverse modeÐless than 10 ms for a 30 V pulse. The normal mode does not require any surface treatment; however, the polymerization is carried out with an in situ electric ®eld enforcing the chiral nematic to unwind and align homeotropically. When after photopolymerization the ®eld is removed, the liquid crystal is in¯uenced by the aligned polymer to stay locally aligned along the polymer network and perpendicular to the surface, while in other areas it relaxes back to the helical structure. A kind of ®ne grain focal conic structure is therefore formed as a result of the competition between intrinsic chirality of the liquid crystal and the frustrating e¨ect of the polymer. The normal mode display is illustrated in Figure 12.33. In the zero-voltage state the material is strongly scattering for all polarizations of incident light and the cell takes on an opaque appearance. When a voltage is applied, the focal conic structure transforms into the homeotropic aligned state and the display is transparent. The underlying function of the polymer in the normal mode display is to create poly-domain focal conic texture and control the domain size. For a 10 mm cell, the normal mode display critical voltage is about 16 V, with a dynamic response of 40 ms to turn on the display and approximately 15 ms to relax

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

421

Figure 12.33. Schematic illustration of a normal mode polymer stabilized display cell. In the zero-voltage state the display scatters light (left), while in the homeotropic texture the display is transparent.

back to the focal conic texture. The normal mode cell does have a signi®cant hysteresis associated with its transmission±voltage curve. The cholesteric nematic that is the most promising in the display industry, and perhaps has received the most attention, is the one that re¯ects light in the visible spectrum. The re¯ective cholesteric display, or simply Ch-LCD, is bistable and exploits the selective Bragg re¯ection properties of cholesteric liquid crystals. Bistability can be brought about by either polymer network stabilization [18], [48] or surface stabilization [49], [50], [51] and, for polymer stabilization, the polymer concentration can be as low as 1% [52], [19] by weight or exceed 20% [54]. A bistable Ch-LCD is schematically depicted in Figure 12.34 with two zero-®eld stable structuresÐthe planar (Figure 12.34(a)) and focal-conic (Figure 12.34(b)). The real planar texture is certainly not perfect, because any perturbation in the cell homogeneity results in some distortion in the pitch axis conformation. Doane and St John have studied the detailed optical characteristics of such imperfect planar textures using di¨use and partially di¨use incident light [50], [51]. Although slightly imperfect such a planar texture represents a bright re¯ection state in accordance with Bragg's law discussed earlier. By application of a su½cient voltage Vfc the planar texture resorts to the focal conic texture. In the focal conic state, the optic axis is ill-de®ned and weakly scatters light. By a½xing an absorbing black layer on the rear of the substrate, the planar texture reveals a bright iridescent re¯ection and the focal conic reveals a black background. The applied voltage can be removed after switching the planar texture to the focal conic texture and the focal conic texture will remain inde®nitely. The path to switch the Ch-LCD back to the planar state through the homeo-

422

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.34. Schematic illustration of a polymer-stabilized cholesteric texture (PSCT) in (a) the planar texture, (b) the focal conic texture, and (c) the homeotropic texture. Transforming the homeotropic aligned state (c) back to the planar of focal conic state depends on the history of the voltage.

Figure 12.35. Data recorded on a yellow re¯ecting Ch-LCD in terms of re¯ection/ contrast (left) and drive voltage (right).

tropic aligned state (Figure 12.34(c)) depends on how the voltage is removed. The liquid crystal can be aligned to the homeotropic state upon application of a voltage VH , which is greater than Vfc . If the applied voltage is abruptly removed, the planar texture is restored, while a slow removal of the ®eld restores the focal conic texture. The electro-optic curves for a polymer stabilized Ch-LCD are shown in Figure 12.35. A large contrast is observed between the planar state and the focal conic texture. In the laboratory, the transformation between the re¯ecting planar state and the focal conic texture is typically achieved by a single ac voltage pulse. A low-voltage pulse …Vfc < V < VH † drives the material into the scattering state and a high-voltage pulse …V > VH † drives the material back into the re¯ecting planar state. The voltage in¯uence of a typical Ch-LCD is shown in Figure 12.35 (right). A single 100 ms ac (1 KHz) pulse is applied to the cell and, after the material responds, its re¯ection is measured. The cell initially in the planar re¯ecting state, eventually trans-

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

423

Figure 12.36. A simple passive drive scheme for bistable PSCT displays.

forms to the focal conic state also stable in the zero ®eld, and then transforms back to the planar state after the material is homeotropically aligned and the voltage is removed abruptly. Doane and coworkers [51] have extensively studied these electro-optic curves as functions of both amplitude and pulse width. The viewing angle, or isocontrast plot, of a Ch-LCD has been shown to be very symmetric [55], which is a necessity for a large number of viewing environments. The attractiveness of bistability and the challenge of how to utilize it in real-world display applications has stimulated many studies on the driving schemes to access this bistable memory. Yang and coworkers [56] have designed simple drive schemes for binary and gray scale displays [57]. Figure 12.36 shows a simple 3 by 3 display to illustrate the driving concept. The voltages given in Figure 12.36 are for a display [56], [57] that will transform into the focal conic state at 35 V and homeotropically align at 50 V as depicted in the ®gure. The row I voltage is either 35 V for the selected row or 0 V for the nonselected row. The column voltage is VR ˆ 15 V for obtaining a re¯ecting state and 0 V for obtaining the scattering focal conic state. The row and column voltages are ac square pulses and have opposite phases. Hence, for the pixels in the selected row the voltage across the pixel is either 50 V to obtain the re¯ecting state or 35 V to obtain the scattering state. For the pixels in the nonselected row the voltage across the pixel is either 0 V or 15 V, under which the states of the pixel remain unchanged. Therefore, once a pixel of a multiplexed display is addressed into the re¯ecting or scattering state, it will remain there inde®nitely. Yang and coworkers [56] have demonstrated impressive prototype displays using this drive scheme. There are several practical attempts to adapt the Ch-LCD to a passive

424

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.37. A simple ac passive drive scheme for bistable PSCTs showing row, column, and pixel waveforms.

matrix, where commercially available driver chips can be used. Pfei¨er and coworkers [58] presented a 14 in diagonal high-resolution Ch-LCD with 100 dots-per-inch (dpi) using commercially available driver chips. The rows were driven with HV6008PG drivers from Supertex while the columns were driven with SED1191F drivers from Seiko Epson. Figure 12.37 shows the drive scheme. The rows are selected with a 20 ms, 250 Hz, G37 V square wave (Vrow ) pulse while the nonselected rows are connected to ground. The data signal, Vcol ˆ G4 V square wave of the same frequency and duration, does not in¯uence the texture in the pixel in the nonselected rows. Pixels are selected with a 180 phase-shifted data signal leading to a select voltage of G41 V. The data signal for the nonselected pixels is in phase with the row signal leading to a nonselect voltage of 33 V. Pfei¨er et al. [58] demonstrated a 1152 by 896 high-resolution pixel display. Most recently, Gandhi and Yang [59] studied the gray scale capability of bistable Ch-LCDs driven by single phase pulses and demonstrated the use of both amplitude modulation and pulse width modulation. Both the polymer network stabilized and surface stabilized Ch-LCD are promising for a number of applications because of their bistability and low power. There is, however, one underlying issue that prevents the technology from being able to switch at video rates. The transition times among the planar, focal conic, and homeotropic aligned textures are long. Huang and coworkers [60], [61] uncovered a rapid transition from the ®eld-induced homeotropic state to the transient planar state through a conical relaxation process on the order of a microsecond. They coined the name ``Dynamic Drive Scheme'' that capitalizes on this transition to allow a substantially

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

425

Figure 12.38. A dynamic drive scheme for bistable PSCTs: (a) three phase; (b) ®ve phase; (c) three phase with amplitude modulation; and (d) three phase with pulse width modulation.

faster addressing speed. They designed a three-phase drive sequence that is schematically illustrated in Figure 12.38(a). The three-phase drive scheme is broken up into distinct stages: preparation, selection, and evolution. The waveform of the voltages is a 100 Hz alternating square wave. In the preparation state, a su½ciently long voltage pulse (60 ms) of amplitude VP aligns the liquid crystal material into the homeotropic state. In the selection state, a pulse VS is set to an appropriate level, either to hold the liquid crystal in the homeotropic aligned state (VON ) or to initiate the transition from the homeotropic state to the transient planar state (VOFF ). The selection pulse is very short (@ 1 ms). In the evolution stage (@ 60 ms), the voltage VE is applied to hold the liquid crystal in the homeotropic state if it was held there by VS or to force the liquid crystal to evolve from the transient planar state to the focal conic state. By the end of the evolution stage, the liquid crystal is in either the homeotropic state or the focal conic state. Upon removal of the evolution voltage, the liquid crystal either transitions from the homeotropic state to the planar state or stays in the focal conic state. The entire drive scheme is approximately 120 ms long, but the ®nal state of the liquid crystal is determined at the selection state which is only 1 ms long. This enables much faster addressing times. The technique has also been recently demonstrated on plastic substrates [62], and adapted to provide gray levels [63]. Further it has been implemented with a unipolar scheme to reduce the overall voltage swing [64]. Mi and Yang [65] recently engaged in a theoretical study to ascertain some of the details behind the homeotropic planar transition using a tensor relaxation method. Zhu and Yang [66] have pushed the envelope of Ch-LCD switching even further with their implementation of a ®ve-phase dynamic drive scheme shown in Figure 12.38(b), enabling a 1000-line display to be addressed in 50 ms. So far we have only discussed the binary operation of a Ch-LCD

426

G.P. Crawford, D. SvensÏek, and S. ZÏumer

sample, but Huang and coworkers [63] recently proposed two novel ways to achieve gray levels with the dynamic drive scheme. These gray-scale drive schemes are shown in Figure 12.38(c) and (d). In the amplitude modulation mode (Figure 12.38(c)), the voltage amplitude of the selection pulse is varied according to the desired level of gray. For example, the higher the selection voltage, the more domains are held in the homeotropic state during the selection time and switched to the planar state after the pulse. The display is therefore brighter at that pixel. Using pulse width modulation, there are only two voltage levels at the selection stage. The duration of VON varies to the desired ®nal level of gray. The longer the duration of VON , the more domains are maintained in the homeotropic state during the selection stage and the higher the ®nal re¯ectance of the display pixel. Huang and coworkers [63] utilized this technique to produce a 133 dpi VGA display with 16 levels of gray. On the ®nal note regarding the dynamic drive scheme, one also needs to take into account a temperature compensation since the transitions between cholesteric textures in Ch-LCDs are temperature-dependent. Gandhi and Yang [67] have investigated the temperature compensation of the dynamic drive scheme so that the Ch-LCDs could operate over a broad temperature range. The passive addressing scheme has seen a lot of attention recently because of its vital role in manufacturing a reliable display. Another area that has seen a lot of activity is adapting these Ch-LCDs to full-color operation. There are essentially two avenues in which one can pursue full color operation in re¯ective Ch-LCDsÐan integrated vertical stack or the more conventional spatial color synthesis. These are schematically shown in Figure 12.39. Both techniques have their pros and cons. Spatial color synthesis can be addressed with current drive schemes but the overall brightness is reduced by a factor of one-third. The integrated vertical stack preserves resolution at the expense of a much more complicated addressing scheme. Chien and coworkers [68] devised a material scheme to manufacture red, green, and blue pixels of polymer-stabilized cholesteric textures by selectively adjusting the chiral pitch length spatially across the display substrate using

Figure 12.39. Two methods to obtain full-color displays.

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

427

Figure 12.40. Data recorded from an integrated RGB stack of cholesteric LCDs, showing the spectral power distribution (left) and the corresponding chromaticity coordinates (right) (courtesy of J.W. Doane).

``photolithography-type'' patterning techniques. Using masking techniques and tunable chiral material, they successfully managed to spatially segregate red, green, and blue re¯ecting pixels by varying the UV dosage across the material. Okada and coworkers [69] and Huang and coworkers [70] have been developing full color displays based on the integrated stacking method. Figure 12.40 presents the spectral power distributions (left) and the corresponding color gamut (right). The overall reported brightness approaches 40%. The color gamut shown in Figure 12.40 is washed out. This can be broadened simply by reducing the width of the spectral power distributions at the expense of the overall brightness. Fabrication of a multicolor display using the stacking method is a di½cult task. In addition to the alignment issues, the drive scheme is complicated, too. Despite all of this, Huang and coworkers [70] have fabricated a full color Ch-LCD with 4096 colors. A photograph of a full-color Ch-LCD display is presented in Figure 12.41. It is 18 VGA (240  160 pixel), 100 dpi, and constructed from a red±green±blue stack of Ch-LCDs.

12.7

Conclusions

Polymer-dispersed chiral liquid crystals and polymer stabilized chiral liquid crystals are very promising materials for ¯at panel display applications allowing us to make thin, adapted to plastic, low-power consumption, lightweight displays particularly useful for numerous portable applications. The application potential of these materials has driven several basic scienti®c studies in this area. The particular area of interest addressed in this chapter is how con®nement can modify the macroscopic and microscopic ordering of chiral nematics. Even in one of the simplest con®ned systems, where a chiral

428

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.41. A photo of a full-color image of a 18 VGA (240  160 pixel) 100 dpi, RGB triple stack. Sixteen levels of gray on each color provide 4096 colors. The display was manufactured by Kent Displays, Inc. (courtesy of J.W. Doane).

liquid crystal is con®ned to droplets, the boundary conditions are fundamentally incompatible with the planar bulk structure of a cholesteric, which results in several fascinating topological defects. Polymer network spread through the liquid crystal adds many more intriguing features to the liquid crystal systems. The polymer network, if polymerized in the chiral liquid crystal phase, mimics the order and orientation of the environment in which it was formed. Desired structures of polymer networks can be achieved by using alignment layers or external ®elds during polymerization. After polymerization the polymer networks can be used in turn to align the chiral liquid crystal. In some instances, the polymer network can be used to achieve a desired director con®guration, which may be impossible to do by surface treatments only. There is a tremendous amount of work to be done in this area in the future both from the applied and basic perspective. We have only seen the beginning of this intellectually rich and challenging ®eld of research. Acknowledgments. This work was supported by the National Science Foundation US-Slovenia grant 9815313, the Ministry of Science and Technology of Slovenia (Grant No J1-0595-1554-98), and the European Union TMR program FMRX-CT98-0209.

References [1] Liquid Crystals in Complex Geometries Formed by Polymer and Porous Networks (edited by G.P. Crawford and S. ZÏumer), Taylor and Francis, London, 1996.

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

429

[2] P.S. Drzaic, Liquid Crystal Dispersions, World Scienti®c, Singapore, 1995. [3] J.W. Doane, N.S. Waz, B.-G. Wu, and S. ZÏumer, Field controlled light scattering from nematic microdroplets, Appl. Phys. Lett. 48, 269 (1986). [4] G.P. Crawford, J.W. Doane, and S. ZÏumer, Polymer dispersed liquid crystals: Nematic droplets and related systems, in: Handbook of Liquid Crystal Research (edited by P.J. Collings and J.S. Patel), Oxford University Press, Oxford, 1997. [5] Y. Bouligand and F. Livolant, The organisation of cholesteric sphereites, J. Physique 45, 1899 (1984). [6] M.V. Kurik and O.D. Lavrentovich, Defects in liquid crystals: Homotopy theory and experimental studies, Uspekhi Fiz. Nauk 154, 381 (1988); (Sov. Phys. Uspekhi 31, 196 (1988)). [7] J. BezicÏ and S. ZÏumer, Structures of the cholesteric liquid crystal droplets with parallel surface anchoring, Liq. Cryst. 11, 593 (1992). [8] J. BezicÏ and S. ZÏumer, Chiral nematic liquid crystals in cylindrical cavities: A classi®cation of planar structures and models of nonsingular disclination lines, Liq. Cryst. 14, 1695 (1993). [9] P.P. Crooker and D.K. Yang, Polymer-dispersed chiral liquid crystal color display, Appl. Phys. Lett. 57, 2529±2531 (1990). [10] H.-S. Kitzerow and P. Crooker, Polymer-dispersed cholesteric liquid crystalsÐ Challenge for research and application, Ferroelectrics 122, 183±196 (1991). [11] H.-S. Kitzerow, Polymer-dispersed and polymer-stabilized chiral liquid crystals, in: Liquid Crystals in Complex Geometries Formed by Polymer and Porous Networks (edited by G.P. Crawford and S. ZÏumer), Taylor and Francis, London, 1996, pp. 187±220. [12] H.-S. Kitzerow, B. Liu, F. Xu, and P.P. Crooker, E¨ect of chirality on liquid crystals in capillary tubes with parallel and perpendicular anchoring, Phys. Rev. E 54, 568 (1996). [13] J. Bajc, J. BezicÂ, and S. ZÏumer, Chiral nematic droplets with tangential anchoring and negative dielectric anisotropy in an electric ®eld, Phys. Rev. E 51, 2176 (1995). [14] J. Bajc and S. ZÏumer, Structural transition in chiral nematic liquid crystal droplets in an electric ®eld, Phys. Rev. E 55, 2925 (1997). [15] M. AmbrozÏicÏ and S. ZÏumer, Chiral nematic liquid crystals in cylndrical cavities, Phys. Rev. E 54, 5187 (1996). [16] M. AmbrozÏicÏ and S. ZÏumer, Axially twisted chiral nematic structures in cylindrical cavities, Phys. Rev. E 59, 4153±4160 (1999). [17] R.A.M. Hikmet, Electrically induced light scattering from anisotropic gels, J. Appl. Phys. 68, 4406 (1990). [18] D.-K. Yang, L.-C. Chien, and J.W. Doane, Cholesteric liquid crystal/polymer dispersion for haze-free light shutters, Appl. Phys. Lett. 60, 3102±3104 (1992). [19] D.-K. Yang, L.-C. Chien, and Y.K. Fung, Polymer stabilized cholesteric textures, Liquid Crystals in Complex Geometries Formed by Polymer and Porous Networks (edited by G.P. Crawford and S. ZÏumer), Taylor and Francis, London, 1996, pp. 103±142. [20] I. Dierking, L.L. Kosbar, A.C. Lowe, and G.A. Held, Polymer network structure and electro-optic performance of polymer stabilized cholesteric textures: I. The in¯uence of curing temperature, Liq. Cryst. 24, 387±395 (1998). [21] I. Dierking, L.L. Kosbar, A.C. Lowe, and G.A. Held, Polymer network structure and electro-optic performance of polymer stabilized cholesteric textures: II. The e¨ect of UV curing conditions, Liq. Cryst. 24, 397±406 (1998).

430

G.P. Crawford, D. SvensÏek, and S. ZÏumer

[22] G.A. Held, L.L. Kosbar, I. Dierking, A.C. Lowe, G. Grinstein, V. Lee, and R.D. Miller, Confocal microscopy study of texture transitions in a polymer stabilized cholesteric liquid crystal, Phys. Rev. Lett. 79, 3443±3446 (1997). [23] G.P. Crawford, Re¯ective liquid crystal display materials: Liquid crystal polymer dispersions, 27th International SAMPE Technical Conference, pp. 484±496, 1995. [24] H.S. Kitzerow and P. Crooker, Behavior of PDLC droplets with negative dielectric anisotropy in electric ®elds, Liq. Cryst. 11, 561±568 (1992). [25] P.G. de Gennes and J. Prost, The Physics of Liquid Crystals, Oxford University Press, Oxford, 1995. [26] L.D. Landau and E.M. Lifshitz, Theory of Elasticity, 3rd ed., Reed Educational, 1986, pp. 144±172. [27] P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, 1997. [28] E.B. Priestley, P.J. Wojtowicz, and Ping Sheng, Introduction to Liquid Crystals, Plenum Press, New York, 1974. [29] G. Vertogen and W.H. de Jeu, Thermotropic Liquid Crystals, Fundamentals, Springer-Verlag, Berlin, 1988. [30] M. KleÂman, Points, Lines and Walls in Liquid Crystals, Magnetic Systems and Various Media, Wiley, New York, 1983. [31] B. JeÂroÃme, Surface e¨ects and anchoring in liquid crystals, Rep. Prog. Phys. 54, 391 (1991). [32] T.J. Sluckin and A. Poniewierski, Novel surface phase transition in nematic liquid crystals: Wetting and the Kosterlitz±Thouless transition, Phys. Rev. Lett. 55, 2907 (1985). [33] M. Nobili and G. Durand, Disorientation-induced disordering at a nematicliquid±crystal-solid interface, Phys. Rev. A 46, R6174 (1992). [34] J. Bajc, Statika in dinamika defektov v ograjenih nematskih tekocÏekristalnih fazah, PhD Thesis, University of Ljubljana, 1998. [35] A. Rapini and M. Papoular, Distorsion d'une lamelle neÂmatique sous champ magneÂtique conditions d'ancrage aux parois, J. Phys. Colloq. France 30, C4±54 (1969). [36] J. Cognard, Alignment of nematic liquid crystals and their mixtures, Mol. Cryst. Liq. Cryst., Suppl. Ser. 1, 1 (1982). [37] S. Kralj and S. ZÏumer, Freedericksz transitions in supra-mm nematic droplets, Phys. Rev. A 45, 2461 (1992). [38] A. Sonnet, A. Kilian, and S. Hess, Alignment tensor versus director: Description of defects in nematic liquid crystals, Phys. Rev. E 52, 718 (1995). [39] N.D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51, 591 (1979). [40] N.D. Mermin, Games to play with 3 He-A, Physica 90, 1 (1977). [41] M.V. Kurik and O.D. Lavrentovich, Defects in liquid crystals: homotopy theory and experimental studies, Uspekhi Fiz. Nauk 154, 381 (1988). [42] D. Monselesan and H.R. Trebin, Temperature dependence of the elastic constants for biaxial nematic liquid crystals, Phys. Stat. Sol. (B) 155, 349 (1989). [43] H.-S. Kitzerow and P.P. Crooker, Electric ®eld e¨ects on the droplet structure in polymer dispersed cholesteric liquid crystals, Liq. Cryst. 13, 31 (1993). [44] G. Barbero, I. Dozov, J.F. Palierne, and G. Durand, Order-electricity and surface orientation in nematic liquid crystals, Phys. Rev. Lett. 50, 2056 (1986). [45] K. Kato, K. Tanaka, S. Tsuru, and S. Sakai, Multipage display using stacked polymer-dispersed liquid crystal ®lms, Jpn. J. Appl. Phys. 32, 4594±4599 (1993);

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

[46] [47] [48] [49] [50] [51] [52] [53] [54] [55]

[56] [57] [58] [59] [60] [61] [62] [63] [64]

431

Color image formation using polymer-dispersed cholesteric liquid crystal, Jpn. J. Appl. Phys. 32, 4600±4604 (1993). K. Kato, K. Tanaka, S. Tsuru, and S. Sakai, Re¯ective color display using polymer-dispersed cholesteric liquid crystal, Jpn. J. Appl. Phys. 33, 2635±2640 (1994). K. Kato, K. Tanaka, S. Tsuru, and S. Sakai, Characteristics of right- and lefthanded polymer-dispersed cholesteric liquid crystal, Jpn. J. Appl. Phys. 33, 4946±4949 (1994). D.-K. Yang, L.-C. Chien, and J.W. Doane, Cholesteric liquid crystal/polymer geldispersion: Re¯ective display application, SID Digest, 759±763 (1992). J.W. Doane, D.-K. Yang, and Z. Yaniv, Front-lit ¯at panel display from polymer stabilized cholesteric textures, Asia Display 92, 73±77 (1992). J.W. Doane and W.D. St John, Re¯ective cholesteric displays, Asia Display 95, 47±50 (1995). B. Taheri, J.W. Doane, D. Davis, and W.D. St John, Optical properties of bistable cholesteric re¯ective displays, SID Digest 96, 39±42 (1996). B.-G. Wu, J. Gao, Y.D. Ma, V. Zhou, and R.B. Timmons, Zero ®eld, multistable cholesteric liquid crystal displays, International Display Research Conference 94, pp. 476±479, 1994. A. Saupe, Disclinations and properties of the director ®eld in nematic and cholesteric liquid crystals, Mol. Cryst. Liq. Cryst. 21, 211±238 (1973). H. Yuan, Bistable re¯ective cholesteric displays, in: Liquid Crystals in Complex Geometries Formed by Polymer and Porous Networks (edited by G.P. Crawford and S. ZÏumer), Taylor and Francis, London, 1996, pp. 265±280. J.L. West, The challenge to new applications to liquid crystal displays, in: Liquid Crystals in Complex Geometries Formed by Polymer and Porous Networks (edited by G.P. Crawford and S. ZÏumer), Taylor and Francis, London, 1996, pp. 255±264. D.-K. Yang, J.W. Doane, Z. Yaniv, and J. Glasser, Cholesteric re¯ective display: Drive scheme and contrast, Appl. Phys. Lett. 64, 1905±1907 (1994). D.-K. Yang and J.W. Doane, Cholesteric liquid crystal/polymer gel dispersion: Re¯ective display application, SID Digest, 759±763 (1992). M. Pfei¨er, D.-K. Yang, J.W. Doane et al., A high-information-content re¯ective cholesteric display, SID Digest 95, 706±709 (1995). J. Gandhi, D.-K. Yang, X.-Y. Huang and N. Miller, Gray scale drive schemes for bistable cholesteric re¯ective displays, Asia Display 98, 127±130 (1998). X.-Y. Huang, D.-K. Yang, P.J. Bos, and J.W. Doane, Dynamic drive for bistable re¯ective cholesteric displays: A rapid addressing scheme, SID Digest 95, 347±350 (1995). X.-Y. Huang, D.-K. Yang, and J.W. Doane, Transient dielectric study of bistable re¯ective cholesteric displays and design of rapid drive scheme, Appl. Phys. Lett. 67, 1211±1213 (1995). G.M. Podojil, D.J. Davis, X.-Y. Huang, N. Miller, and J.W. Doane, Plastic VGA re¯ective cholesteric LCDs with dynamic drive, SID Digest 98, 51±54 (1998). X.-Y. Huang, N. Miller, A. Khan, D. Davis, J.W. Doane, and D.-K. Yang, Gray scale of bistable re¯ective cholesteric displays, SID Digest 98, 810±813 (1998). X.-Y. Huang, N. Miller, and J.W. Doane, Unipolar implementation for the dynamic drive scheme of bistable re¯ective cholesteric displays, SID Digest 97, 899±902 (1997).

432

G.P. Crawford, D. SvensÏek, and S. ZÏumer

[65] X.-D. Mi and D.-K. Yang, Cell designs for fast re¯ective cholesteric LCDs, SID Digest 98, 909±912 (1998). [66] Y.-M. Zhu and D.-K. Yang, High-speed dynamic drive scheme for bistable re¯etive cholesteric displays, SID Digest 97, 97±100 (1997). [67] J. Gandhi and D.-K. Yang, Temperature compensation of the dynamic drive scheme for bistable re¯ective cholesteric displays, SID Digest 98, 794±797 (1998). [68] L.-C. Chien, U. Muller, M.-F. Nabor, and J.W. Doane, Multicolor re¯ective cholesteric displays, SID Digest 95, 169±171 (1995). [69] M. Okada, T. Hatano, and K. Hashimoto, Re¯ective multicolor display using cholesteric liquid crystals, SID Digest 97, 1019±1023 (1997). [70] X.-Y. Huang, A. Khan, D. Davis, C. Jones, N. Miller, and J.W. Doane, Full color (4096 colors) re¯ective cholesteric liquid crystal display, Asia Display 98, 883±886 (1998). [71] G.P. Crawford, R.D. Polak, A. Scharkowski, L.-C. Chien, S. ZÏumer, and J.W. Doane, Nematic director-®elds captured in polymer networks con®ned to spherical droplets, J. Appl. Phys. 75, 1968 (1994). [72] G.P. Crawford, A. Scharkowski, Y.K. Fung, J.W. Doane, and S. ZÏumer, Internal surface, orientational order, and distribution of a polymer network in a liquid crystal matrix, Phys. Rev. E 52, R1273±76 (1995). [73] A. Jakli, L. Bata, K. Fodor-Csorba, L.I. Rosta, and L. Noirez, Structure of polymer networks dispersed in liquid crystals: small angle neutron scattering study, Liq. Cryst. 17, 227 (1994). [74] Y.K. Fung, A. BorsÏtnik, S. ZÏumer, D.-K. Yang, and J.W. Doane, Pretransitional nematic ordering in liquid crystals with dispersed polymer networks, Phys. Rev. E 55, 1637±1645 (1997). [75] A. Golemme, S. ZÏumer, D.W. Allender, and J.W. Doane, Continuous nematicisotropic transition in submicron-size liquid-crystal droplets, Phys. Rev. Lett. 61, 1937 (1988). [76] G.P. Crawford, R. Ondris-Crawford, S. ZÏumer, and J.W. Doane, Anchoring and orientational wetting transitions of con®ned liquid crystals, Phys. Rev. Lett. 70, 1838 (1993). [77] T. Bellini, N.A. Clark, D.D. Muzny, L. Wu, C.W. Garland, D.W. Schaefer, and B.J. Oliver, Phase behavior of the liquid crystal 8CB in a silica aerogel, Phys. Rev. Lett. 69, 788 (1992). [78] G.S. Iannacchione, G.P. Crawford, S. ZÏumer, J.W. Doane, and D. Finotello, Randomly constrained orientational order in porous glass, Phys. Rev. Lett. 71, 2595 (1993). [79] A. Jakli, D.R. Kim, L.-C. Chien, and A. Saupe, E¨ect of a polymer network on the alignment and the rotational viscosity of a nematic liquid crystal, J. Appl. Phys. 72, 3161 (1992). [80] M. Vilfan, G. Lahajnar, I. ZÏupancÏicÏ, S. ZÏumer, R. Blinc, G.P. Crawford, and J.W. Doane, Dynamics of a nematic liquid crystal constrained by a polymer network: a proton NMR study, J. Chem. Phys. 103 (1995). [81] Y.K. Fung, D.-K. Yang, S. Ying, L.-C. Chien, S. ZÏumer, and J.W. Doane, Polymer networks formed in liquid crystals, Liq. Cryst. 15, 797±801 (1996). [82] S. ZÏumer, A. Golemme, and J.W. Doane, Light extinction in a dispersion of small nematic droplets, J. Opt. Soc. Amer. A 6, 403 (1989). [83] S. ZÏumer, Light scattering from nematic droplets: Anomalous di¨raction approach, Phys. Rev. A 37, 4006±4015 (1988).

13

Chirality in Liquid Crystal Elastomers P. Stein and H. Finkelmann

13.1

Introduction

Liquid crystal elastomers combine the unique properties of rubber elasticity with the anisotropy of the liquid crystalline state. Although intensive research has been performed during the past few years, and the ®rst liquid crystal elastomers had only been synthesized in 1981, knowledge about these materials is still in its infancy. This is due to the complex interplay between the polymer network and the anisotropic order of the liquid crystalline state. On the other hand, liquid crystal elastomers o¨er new aspects for chemistry, physics, and material science [1], [2], [3], [4]. The basic chemistry of liquid crystal elastomers is very similar to the chemistry of linear liquid crystal polymers. One can distinguish between main- and side-chain elastomers where the mesogenic units are either segments or side groups of the monomer units of the macromolecular chain. The elastomers only di¨er from linear liquid crystal polymers by conventional crosslinking reactions. New aspects for chemistry appear when liquid crystal networks are required that spontaneously, and without any external ®elds, exhibit a macroscopically uniform orientation of the director. For these conditions, networks have to be sythesized where the chain conformation is consistent a priori with the uniform liquid crystal order [5]. New physical aspects arise due to the coupling between the network chains and the anisotropy of the liquid crystal state [6], [7]. External mechanical ®elds directly a¨ect the network chain conformation that interacts with the liquid crystal order. In contrast to enthalpy elasticity of solid-state material, the rubber elasticity, or entropy elasticity of the networks allows extensive deformations while the mechanical moduli may be smaller by some orders of magnitude. In this way liquid crystal elastomers are ``soft crystals'' o¨ering numerous aspects for application in material science. The chirality in liquid crystal elastomers can be at the origin of additional physical properties such as ferroelectricity, pyroelectricity, circular dichroism, and nonlinear optics coupled to the polymer network. Applying external mechanical ®elds to the elastomers consequently causes electro433

434

P. Stein and H. Finkelmann

mechanical and optomechanical e¨ects that do not exist in conventional low molar mass liquid crystals or conventional isotropic polymer networks. During the past few years, some basic experimental and theoretical investigations on chiral nematic (cholesteric) and chiral smectic C phases have been performed. In this chapter we will brie¯y summarize these results.

13.2

Cholesteric Elastomers

In 1989, Brand [8] published a theoretical model that predicts electromechanical e¨ects in cholesteric elastomers. A compression of the helicoidal axis should result in a polarization. Four years later, Terentjev [9] published a theory about piezoelectricity in cholesteric elastomers. He also came to the conclusion that a compression along the helicoidal axis produces polarization. But in 1995, Pelcovits and Meyer [10] remarked that the theory of Brand does not include the centrosymmetric character of the helicoidal superstructure. Brand described the cholesteric helix by a vector, which produces a polar character. Concerning Terentjev's theory, Pelcovits et al. criticized an insu½cient coarse graining in the model. In their model, a compression along the helicoidal axis does not produce a polarization, while a shear force should give an electromechanical response. The controversial discussions about the di¨erent theoretical models stimulated experimental investigations on cholesteric elastomers. In the next sections, di¨erent synthetical approaches are summarized for cholesteric elastomers. Thereafter, electromechanical investigations are reviewed.

13.2.1

Preparation

The ®rst cholesteric elastomers were synthesized in 1981 on the basis of liquid crystal side-chain polymers by a hydrosilylation reaction of chiral mesogenic groups to poly(methylhydrogensiloxane) [11]. For these chiral networks, only the synthesis and the liquid crystal phase behavior was described. In 1988, Zentel described the preparation of liquid crystal elastomers with chiral mesogens [12], [13]. Chiral liquid crystal phases were obtained by the cross-linking reaction of combined main- and side-chain prepolymers having free vinyl groups. The prepolymers were cross-linked via a hydrosilylation reaction with oligo(methylhydrogensiloxane). The elastomers exhibit cholesteric, and for the ®rst time chiral smectic C phases. By applying a uniaxial mechanical stress to the networks, an unwinding of the cholesteric pitch was observed. Another method to obtain cholesteric elastomers was applied by Meier [14]. The starting material is a nematic network that is swollen by a chiral low molar mass liquid crystal. Similar to conventional induced cholesteric phases, the concentration of the chiral dopant (or swelling solvent) determines the cholesteric pitch within the network. A linear relationship between

13. Chirality in Liquid Crystal Elastomers

a)

435

b)

Figure 13.1. Cholesteric elastomer: the initially unloaded sample exhibits a polydomain and is turbid (1a). Compression causes a transition into a translucent monodomain (1b) (reproduced with permission from [14]).

the pitch p and the concentration of the chiral swelling solvent exists. In the inital state, the networks are macroscopically nonordered with respect to the director (polydomain, SD ˆ 0; SD ˆ director order parameter). They strongly scatter light and are completely turbid. Applying a uniaxial compression force to the nonordered network causes a director orientation with the helicoidal axis parallel to the compression force. At a certain threshold deformation lth or threshold stress, a monodomain exists that optically behaves like a single crystal of an organic or inorganic material. The network is completely translucent as seen in Figure 13.1(b). This mechanicallyinduced polydomain/monodomain transition is reversible. Releasing the stress or strain causes the transition into the initial nonordered state. Above lth , the wavelength of re¯ection lR of the network is linearly dependent on the deformation, as shown in Figure 13.2 for two networks that di¨er in the concentration of the chiral dopant. From these measurements, it follows that, within experimental error, the pitch of the cholesteric phase a½nely deforms with the macroscopic dimensions of the network. Another method to obtain a uniformly-ordered cholesteric network was applied by Vallerian et al. [15]. They started with a linear cholesteric sidechain polymer which was prepared between glass slides. By surface e¨ects, a Grandjean texture becomes stable. Because the linear polymer additionally contains polymerizable groups, a thermally-induced cross-linking reaction forms the network. A disadvantage of this method is the limited size of the sample and problems in analyzing the mechanical properties of the network. The approach of Chang et al. [16] is very similar to the method described by Vallerian. Instead of starting with a linear liquid crystal polymer, they use

436

P. Stein and H. Finkelmann

Figure 13.2. Wavelength of the re¯ection lR of two cholesteric elastomers as a function of uniaxial compression (reduced temperature Tred ˆ 0:956). The vertical dashed line indicates the threshold deformation lm . For l < 0:72, lR decreases nearly a½ne with the macroscopic network dimensions (reproduced with permission from [18]).

a mixture of low molar mass derivatives in which the nematic component is the functional monomer. With a cross-linking agent and a chiral dopant, the elastomer is synthesized between the glass slides which provide a uniform orientation of the monomer mixture. The polymerization process does not destroy the initial order.

13.2.2

Properties

Electromechanical experiments on cholesteric elastomers carried out by Meier in 1990 [17] seemed to con®rm Brand's theoretical model. On compression of uniformly ordered networks along the helicoidal axis, an electric

13. Chirality in Liquid Crystal Elastomers

437

response was observed. But in 1993, it was mentioned that this electromechanical response was dependent on the shape of the sample [18]. Additionally, the compression causes shear e¨ects if a barrel-like deformation, rather than a cylindrical one, of the sample occurs. Actually, eliminating these experimental di½culties, T. Eckert did not observe an electro-mechanical response of the cholesteric networks under compression in the direction of the cholesteric helix within experimental error [19]. Essentially the same results were obtained by Vallerien [15]. Applying the well-established experimental setup [20] of a piezotranslator-driven dynamical deformation, only a very weak signal could be observed under compression in the direction of the helicoidal axis. In the vicinity of the cholesteric to isotropic phase transformation, the response was above experimental error. This e¨ect might be due to some reorientation of the helicoidal axis resulting in shear deformations. In 1997, Chang et al. provided theoretical and experimental evidence that only a shear ®eld produces piezoelectricity in cholesteric elastomers [16]. Applying a dynamic shear force perpendicular to the helicoidal axis, a nearly linear response is found between the piezoelectric voltage and the shear amplitude, as shown in Figure 13.3 for two frequencies. Only at larger shear displacement, the induced voltage tends toward saturation.

Figure 13.3. Induced voltage from piezoelectricity as a function of the shear displacement with two frequencies produced by a cholesteric elastomer gel. The induced voltage saturates at larger shear displacement (reproduced with permission from [16]).

438

P. Stein and H. Finkelmann

13.3

Chiral Smectic C Elastomers

The occurrence of ferroelectricity in the chiral smectic C phase was already predicted by Meyer in 1975 [21]. From symmetry arguments, he concluded a polarization of a smectic layer perpendicular to the layer normal and to the director. Unwinding of the helicoidal structure causes macroscopic ferroelectric properties of the SC phase [22], [23]. Unlike low molar mass liquid crystals or linear liquid crystal polymers, the unwinding of the helicoidal structure is performed by applying an electric ®eld or by surface e¨ects. For SC elastomers, a mechanical ®eld should cause orientation e¨ects toward uniform alignment of the phase structure. Additionally, the piezoelectric behavior is expected to be similar to the piezoelectricity of solid-state material because the network structure prevents ¯ow. The electro-mechanical behavior of smectic C  networks was already addressed by Brand in 1989 [8]. In agreement with the monoclinic symmetry of these systems, he derived ten piezoconstants for the Sc networks. Furthermore, rotatoelectric e¨ects are predicted. Due to the noncentrosymmetric character of the untwisted SC phase, further nonlinear optical e¨ects like frequency doubling occur.

13.3.1

Preparation

The ®rst chiral smectic C  elastomer was synthesized by Zentel et al. in 1988 [13], [24]. The preparation of the network is similar to the synthesis of the cholesteric networks described above. The SC network was uniaxially stretched. An unwinding of the helicoidal superstructure is observed and an orientation of the director, as well as of the smectic layer normal roughly parallel to the stress axis occurs. A macroscopically uniform alignment of the sample was not observed as explained below. Similar to the techniques described for ordered chiral nematic elastomers, ordered chiral SC elastomers can be obtained either by surface-induced alignment of a functional linear SC pre-polymer (Vallerian et al. [15]) or by electric ®eld-induced alignment of the prepolymer (Brehmer et al. [25]). A method that is not limited by the size of the sample makes use of the mechanical ®eld e¨ects of liquid crystal elastomers [26], [27]. The SC networks are synthesized in a two-step cross-linking reaction. In the ®rst step, a weakly cross-linked gel is prepared and uniaxially loaded. This causes a uniform orientation of the director of the mesogens, but a conical distribution of the smectic layer normal with the tilt angle y to the director. A second deformation at an angle …90 ÿ y† to the director aligns the smectic layers and perfect monodomains are obtained with a bookshelf alignment. Mauzac et al. [28] oriented their networks at the isotropic to liquid crystalline phase transformation by applying an electric poling ®eld [28]. However, the bookshelf alignment is not perfect. Finally Gebhard and Zentel [29] realized ultrathin free-standing ferroelectric ®lms using a method of Skarp [30]. The prepolymer was spin-coated on an NaCl plate and photo-cross-

13. Chirality in Liquid Crystal Elastomers

439

Figure 13.4. Electro-optical investigations at 2 Hz on a chiral smectic C elastomer. An unsymmetrical switching behavior is observed due to the network conformation (reproduced with permission from [25]).

linked. The free-standing ®lm is obtained simply by dissolving the NaCl in water. With this method homeotropically aligned, networks are prepared.

13.3.2

Properties

Electric ®eld e¨ects were investigated in detail by Brehmer et al. [25], [31], and by Zentel and Brehmer on aligned SC elastomers. The electro-optical response strongly increases with the increasing cross-linking density of the network. Additionally, they observed a nonsymmetrical optical response (Figure 13.4). The explanation of this behavior is given by the memory e¨ect of the network. When the electric ®eld is applied in the initial state, the reorientation of the mesogenic groups also causes a reorganization of the network strands which costs elastic energy. The reverse e¨ect into the initial orientation gains elastic energy and is strongly accelerated. For the ultrathin ®lm, Gebhard et al. remarked that the threshold ®eld strength was reduced compared to the ®eld strength in commercial EHC cells. They speculate that this e¨ect might be due to the second missing surface [29]. These ®lms were additionally characterized by dielectric spectroscopy [33]. Nonlinear optical measurements on SC elastomers were performed by J. BenneÁ et al. with mechanically aligned samples. Due to the alignment procedure, the helicoidal superstructure is untwisted and the two-step crosslinking process locks in the ferroelectric structure. Without applying external ®elds, these networks spontaneously produce frequency doubling and, from the SHG signal in a Marker-fringe experiment, the C2 symmetry of the SC elastomer is con®rmed [35], [36].

440

P. Stein and H. Finkelmann

a)

b)

Figure 13.5. Temperature-dependent pyroelectric investigations on a chiral smectic C elastomer. Two di¨erent cross-linking states were measured. (a): 0% cross-linking (polymeric state), (b) 5% cross-linking (transition temperatures: (a) g ÿ 26  C c 18  C SC 85  C i; (b) g ÿ 28  C c 23  C SC 77  C i) (reproduced with permission from [28]).

Mauzac et al. measured the pyroelectric properties of their networks compared to the corresponding linear polymer (Figure 13.5) and calculated the spontaneous polarization. Although the chemistry of the linear polymer is similar to the chemistry of the network, the pyroelectric response of the network is lower. The only conclusion for this result is the nonperfect orientation of the network [28].

13. Chirality in Liquid Crystal Elastomers

441

Figure 13.6. Piezoelectric signal of a chiral smectic C elastomer as a function of the applied dynamic deformation at 35  C …h†, 39  C …y†, and 52  C …r† (transition temperatures: g 1  C SC 81  C i) (reproduced with permission from [15]).

The phase structure of the SC phase is at the origin of the piezoelectric e¨ects. While low molar mass SC liquid crystals ¯ow under the in¯uence of an external mechanical ®eld, the network structure of the SC elastomers prevents macro-Brownian motions of the mesogens and deformations with large amplitudes are feasible. On the other hand, compared to solid-state crystals, the modulus of the elastomers is smaller by orders of magnitude and, moreover, can be modi®ed by the cross-linking density of the network. With these exceptional properties, SC elastomers o¨er a new class of electromechanical materials that stimulate theoretical and experimental activities. Electromechanical measurements were ®rst performed by Vallerian et al. in 1990 [15]. An ordered and surface stabilized prepolymer was cross-linked in a conventional TD-covered glass cell and the mechanical deformation achieved by an oscillating piezotransducer. A linear increase in the piezoelectric signal is observed, with an increasing amplitude of the deformation as shown in Figure 13.6. As expected, the piezoelectric response decreases with the temperature, which can be related to the order parameter and the tilt angle. However, with these measurements, no mechanical stress was analyzed. In 1994, Brehmer et al. [31] additionally analyzed the stress/strain relation that enables quantitative analysis of the electromechanical response as a function of temperature as shown in Figure 13.7. The piezoelectric coe½cient exhibits a maximum at low temperatures of about 6.5 pC/N. Due to the experimental setup, the modulus could only be determined halfquantitatively. It has to be noted that the SA phase …T > 53  C† also clearly

442

P. Stein and H. Finkelmann

Figure 13.7. Piezoelectric signal of a chiral smectic C elastomer as a function of the temperature (transition temperatures: g SX 29  C SC 53  C SA 88  C i) (reproduced with permission from [25]).

exhibits an electromechanical response as theoretically predicted in the same year by Terentjev and Warner [37]. The networks synthesized by Mauzac under the action of an electric poling ®eld also exhibit piezoelectricity. Obviously, due to the nonperfect alignment of the sample, the largest response was about ca. 0.75 pC/N [28]. The advantage of the monodomain networks prepared by Semmler [26], [27], is the macroscopic size of the material that enables careful analysis of the mechanical properties. In 1996, T. Eckert also measured a linear electromechanical response in a static experimental setup [38]. A response of about 3 pC/N was observed for this material. The static experimental setup allows a simple experiment to prove the polar character of the uniformly ordered sample as shown in Figure 13.8. Simply by turning over the elastomers, the sign of the piezoelectric response changes (Figure 13.8, sample 2). The same sample that is only aligned with respect to the director by the uniaxial mechanical ®eld exhibits no response (Figure 13.8, sample 1) and proves the conical alignment of the smectic layer normal to the director. The inverse electromechanical properties of these samples were investigated in detail by Keck using a Michelson interferrometer. Similar to Eckert's result, a linear response is observed at about 10 pm/V (Figure 13.9). Applying additionally a dc bias ®eld enhances the electromechanical response [38], [39]. But this

13. Chirality in Liquid Crystal Elastomers

443

Figure 13.8. Piezoelectric signal of a chiral smectic C elastomer as a function of the applied force. Sample 1: uniaxially deformed SC network. Sample 2: biaxially deformed SC elastomer with C2 symmetry. The existence of the polar axis was proved by turning the sample over (reproduced with permission from [38]).

Figure 13.9. Inverse piezoelectric signal of a chiral smectic C elastomer at 63 Hz as a function of the amplitude of the applied electric voltage, …d† without bias ®eld, …v† with bias ®eld of 40 V (reproduced with permission from [38]).

444

P. Stein and H. Finkelmann

Figure 13.10. Mechanical and electromechanical response of a chiral smectic C elastomer as a function of the temperature for four di¨erent frequencies, …y† 0.158 Hz, …r† 1.12rHz, …h† 11.2 Hz, …i† 100 Hz (transition temperatures: g SX 308 K SC 333 K SA 346 K i). (a) E 0 : mechanical storage modulus, E 00 : mechanical loss modulus; (b) Qs0 : electromechanical storage coe½cient, Qs00 : electromechanical loss coe½cient; (c) Qe0 : electromechanical storage modulus, Qe00 : electromechanic storage modulus.

e¨ect of the dc bias ®eld can be reversed as shown by Lehmann et al. [40]. Very recently, Stein et al. analyzed in detail the network of Eckert. Instead of a static experimental setup, direct dynamic electromechanical measurements were performed [41]. Some of these results are summarized in Figure 13.10. The real and imaginary part of E (E 0 ˆ storage modulus, E 00 ˆ loss

13. Chirality in Liquid Crystal Elastomers

445

modulus) re¯ect the mechanical properties of the network as a function of temperature. The electromechanical response is shown for Qs0 (resp. Qs00 ) related to the mechanical stress, and Qe0 (resp. Qe00 ) related to the mechanical strain. The results are presented for four di¨erent frequencies. Interestingly, Qs0 and Qe0 change sign as a function of temperature; this might be due to conformational variations of the chiral monomer units. Furthermore, it has to be noted that the SA phase also exhibits an electromechanical response. The largest e¨ect of Qs0 is about 60 pC/N (not shown here) although, the chemical constitution of the SC network is absolutely not optimized for electromechanical applications. The SC to SA phase transformation is not re¯ected in a discontinuity of Qs0 or Qe0 at the phase transformation temperature. On the other hand, the NLO measurement on the same system indicates that this phase transformation occurs where the SHG signal vanishes in the SA phase. Finally, the results of Brehmer et al. [42] have to be mentioned. Their interpretation of the electromechanical response in the SA networks is a mechanoclinic e¨ect. The mechanical ®eld, which induces the mechanoclinic e¨ect, is induced by the network, which was formed in the SC phase. This induces a tilt and therefore an electromechanical response.

13.4

Conclusion

The coupling between the properties of conventional polymer networks and the properties of chiral liquid crystalline phases results in interesting, new opto- and electromechanical e¨ects of the chiral liquid crystalline elastomers, as demonstrated by theoretical considerations and experiments. Knowledge about these new materials is still in its infancy. But the properties analyzed so far for these elastomers indicate promising aspects for application and are the basis for the new syntheses of optimized chiral liquid crystal networks.

References [1] H.R. Brand and H. Finkelmann, in: Handbook of Liquid Crystals (edited by D. Demus, J. Goodby, G.W. Gray, H.-W. Spiess, and V. Vill), Vol. 3, pp. 279±302, Wiley-VCH, Weinheim, 1998. [2] S.M. Kelly, Liq. Cryst. 24(1), 71 (1998). [3] F.J. Davis, J. Mater. Chem. 3(6), 551 (1993). [4] R. Zentel, Adv. Mater. 101(10), 1437 (1989). [5] J.C. Dubois, P. Le Barny, M. Mauzac, and C. Noel, Acta Polymer 48, 47 (1997). [6] W. Gleim and H. Finkelmann, Makromol. Chem. 188, 1489 (1987). [7] J. SchaÈtzle, W. Kaufhold, and H. Finkelmann, Makromol. Chem. 190, 3269 (1987). [8] H. Brand, Makromol. Chem., Rapid Commun. 10, 441 (1989). [9] E. Terentjev, Europhys. Lett. 23, 27 (1993). [10] R.A. Pelcovits and R.B. Meyer, J. Phys. II France 5, 877 (1995).

446

P. Stein and H. Finkelmann

[11] H. Finkelmann, H.-J. Kock, and G. Rehage, Makromol. Chem., Rapid Commun. 2, 317 (1981). [12] R. Zentel, Liq. Cryst. 3(4), 531 (1988). [13] R. Zentel, G. Reckert, S. Bualek, and H. Kapitza, Makromol. Chem. 190, 2869 (1989). [14] W. Meier and H. Finkelmann, Mater. Res. Soc. Bull. 16(1), 29 (1991). [15] S.U. Vallerien, R. Zentel, F. Kremer, H. Kapitza, and E.W. Fischer, Makromol. Chem., Rapid Commun. 11, 593 (1990). [16] C.-C. Chang, L.-C. Chien, and R.B. Meyer, Phys. Rev. E 55(1), 534 (1997). [17] W. Meier and H. Finkelmann, Makromol. Chem., Rapid Commun. 11, 599 (1990). [18] W. Meier and H. Finkelmann, Macromolecules 26, 1811 (1993). [19] T. Eckert, PhD Theses, Freiburg, 1995. [20] R. Hayakawa and Y. Wada, Fortschr. Hochpolym.-Forsch. 11, 1 (1973). [21] R.B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, J. Phys. Lett. 36, L-69 (1975). [22] G. Pelzl, in: Liquid Crystals (edited by H. Stegemeyer), Vol. 88, Steinkop¨ Darmstadt, Springer-Verlag, New York, 1994. [23] S. Chandrasekhar, in: Liquid Crystals, Vol. 378, 2nd ed., Cambridge University Press, Cambridge, 1992. [24] R. Zentel, Liq. Cryst. 3(4), 531 (1988). [25] M. Brehmer, R. Zentel, G. Wagenblast, and K. Siemensmeyer, Macromol. Chem. Phys. 195, 1891 (1994). [26] K. Semmler and H. Finkelmann, Polym. Adv. Technol. 5, 231 (1994). [27] K. Semmler and H. Finkelmann, Macromol. Chem. Phys. 196, 3197 (1995). [28] M. Mauzac, H.-T. Nguyen, F.-G. Tournilhac, and S.-V. Yablonsky, Chem. Phys. Lett. 240, 461 (1995). [29] E. Gebhard and R. Zentel, Macromol. Rapid Commun. 19, 341 (1998). [30] K. Skarp, S. Utho, K. Myoijn, H. Moritake, M. Ozaki, B. Helgree, and K. Yoshino, Jpn. J. Appl. Phys. 34, 5433 (1995). [31] M. Brehmer, A. Wiesemann, R. Zentel, K. Siemensmeyer, and G. Wagenblast, Polym. Preprints (Am. Chem. Soc., Div. Polm. Chem.) 34(2), 708 (1993). [32] R. Zentel and M. Brehmer, Adv. Mater. 6(7/8), 598 (1994). [33] E. Gebhard and R. Zentel, submitted to Liquid Crystals. [34] S.V. Shilov, H. Skupin, F. Kremer, K. Skarp, P. Stein, and H. Finkelmann, SPIE 3318, 62 (1998). [35] I. Benne and H. Finkelmann, Macromol. Rapid Commun. 15, 295 (1994). [36] I. BenneÂ, K. Semmler, and H. Finkelmann, Macromolecules 28, 1854 (1995). [37] E.M. Terentjev and M. Warner, J. Phys. II France 4, 11 (1994). [38] T. Eckert, H. Finkelmann, M. Keck, W. Lehmann, and F. Kremer, Macromol. Rapid Commun. 17, 767 (1996). [39] W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, Ferroelectrics 208±209, 373 (1998). [40] F. Kremer, W. Lehmann, H. Skupin, L. Hartmann, P. Stein, and H. Finkelmann, Polym. Adv. Techn. 9, 672 (1998). [41] P. Stein and H. Finkelmann, in preparation. [42] M. Brehmer, R. Zentel, F. Gieûelmann, R. Germer, and P. Zugenmaier, Liq. Cryst. 21, 589 (1996).

14

Phase Chirality of Micellar Lyotropic Liquid Crystals Karl Hiltrop

14.1

Lyotropic Liquid Crystals

The mesogenic units of liquid crystals have to be anisometric (i.e., nonspherical) objects in order to allow for the essential long-range orientational order. In the case of the thermotropic liquid crystals this precondition can be ful®lled in one-component systems; single rod- or disk-like molecules of low molecular weight and also polymers can be suitable. Further possibilities arise if two- or more-component systems are considered and one of the components acts as a solvent for the other; if the variation of solvent concentration leads to phase transitions the system is called lyotropic. As an example, colloidal particles in aqueous suspensions can adopt an orientational long-range order, usually forming achiral phases [1]. Even biomolecules like the tobacco mosaic virus with its long rod-like shape (length about 300 nm, diameter 18 nm) are suited for lyotropic liquid crystalline structures [2]. In solutions of amphiphilic, as well as of certain aromatic matter (e.g., dyestu¨s), aggregates of monomers can form; di¨erent anisometric shapes can be developed and thus the aggregates take the role of the building blocks of lyotropic liquid crystals. A very remarkable example of a nonmicellar chiral system based on an achiral host phase of column-like aggregates of large, disk-like molecules has been published by Usuol'tseva et al. [3]. The disk-like tetranuclear palladium organyls form charge transfer complexes with a suitable chiral electron acceptor in apolar organic solvents. Polymers, e.g., polypeptides, form chiral lyotropic liquid crystalline phases; T. and E. Samulski calculated the mutual van der Waals±Lifshitz forces between rod-like particles with chiral polarizability in an isotropic dielectric medium and therewith explained the phase chirality of polypeptides in organic solvents like dioxane. Furthermore, they showed that the use of di¨erent solvents can cause di¨erent handedness of the helix for the same solutes and that suitable mixtures of such solvents can result in compensated (i.e., nematic) mixtures although the polypeptide is chiral [4]. Solvent-

447

448

K. Hiltrop

Figure 14.1. (a) Schematic structure of an amphiphilic molecule. (b) Assimilation of amphiphilic molecules at interfaces.

induced helix inversions have also been reported for cellulose derivatives in acetic acid/dichloroacetic acid mixtures [5]. However, the main part of this chapter is restricted to micellar LLC systems. Some basic aspects are brie¯y described in Sections 14.1.1 to 14.1.4.

14.1.1

Amphiphilic Matter

Molecules composed of hydrophilic as well as of hydrophobic parts are called amphiphilic, soaps are simple examples. In strongly polar (e.g., water) or apolar solvents (e.g., alkanes) their solubility as monodisperse molecules is small. The contact of the respective lyophobic moieties to the solvent can be avoided by accumulation at the interfaces of the solution, see Figure 14.1. The interfacial properties of solutions are determined distinctly by dissolved amphiphilic matter which for that reason is termed surfactant (surface active agent). Typical examples of ionic (cationic or anionic), zwitterionic, and nonionic surfactants of low molecular weight are sketched in Figure 14.2.

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

Figure 14.2. Structures of di¨erent amphiphiles and typical examples.

449

450

K. Hiltrop

14.1.2

Aggregation

Water is a usual liquid. Nevertheless, it has peculiar properties. Water serves as the almost exclusive standard solvent in lyotropic systems and the following considerations will be restricted to it mostly. Water is polar and dissolves polar matter. Its speciality consists in the capability to form strong intermolecular hydrogen bonds (binding enthalpy about 1 to 50 kJ/mol). The result is the cage-like microstructure of ice (clathrate); these cages partly keep on existing above the melting point of ice in liquid water. Consequently, the dissolution of unpolar molecules which cannot form hydrogen bonds to water requires the provision of a large amount of free enthalpy to break up the clathrates and is unfavorable. The major contribution to this free enthalpy stems from a loss of entropy due to the loss of hydrogen bonds. Overall, this phenomenon is called the hydrophobic e¨ect [6]. Large hydrophobic molecules cannot be solubilized monodispersely in water in considerable concentration. The situation changes if amphiphilic, instead of merely hydrophobic, matter is mixed with water. Then a distinct possibility arises to circumvent the disadvantage of cutting too many hydrogen bonds. Amphiphilic molecules can aggregate in a way that their hydrophobic tails form a microdroplet of alkane surrounded by a shell of hydrophilic headgroups in contact with water, see Figure 14.3. It is the avoidance of intermolecular hydrophilic/hydrophobic contacts combined with the maintenance of attractive interaction between hydrophilic

Figure 14.3. Reversible aggregation of surfactants in a solution resulting in a micelle.

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

451

Figure 14.4. (a) The one-dimensional structure element of aggregation: a tail-to-tail arrangement of two surfactants. (b) and (c) Formation of isometric and anisometric micelles.

headgroups and water that favors the formation of aggregates, the so-called micelles. Micellization starts at a certain so-called critical micelle concentration (CMC). Increasing surfactant concentration leads to micelle growth and therewith to the development of anisometric shapes of the aggregates, like rods or disks, see Figure 14.4. The basic unit is the tail-to-tail arrangement of two amphiphiles which can be developed into two and three dimensions to form isometric and ®nally anisometric units. Thus one essential precondition for the formation of orientational order is achieved. The packing parameter concept of Israelachvili [7] explains the formation of di¨erent shapes of amphiphilic aggregates via the spontaneous curvature of its interface, as a consequence of the space requirements of hydrophobic and hydrophilic moieties of the surfactant molecules and solubilized additives (i.e., dopants), see Figure 14.5. Bulky headgroups lead to strong curvature of the surface of normal micelles, spherical or nearly spherical aggregates will form. Similar cross sections of headgroup and hydrophobic moiety favor plane lamellae. A disk-like micelle is enforced to grow if a solubilized dopant possesses a small headgroup, as is the case for alcohols.

452

K. Hiltrop

Figure 14.5. The shape of an aggregate as the consequence of the shape of the monomers: the concept of the packing parameter, after [7].

14.1.3

Liquid Crystalline Phases, Phase Diagrams

As there is a lack of detailed phase diagrams exhibiting lyotropic phase chirality it may be su½cient to discuss brie¯y the polymorphism of two frequently used achiral systems. The ®rst example is presented by the cationic surfactant hexadecyltrimethylammonium bromide (CTAB) in a binary mix-

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

453

(a) Figure 14.6. Phase diagrams of lyotropic mixtures (temperature versus amphiphile concentration). (a) Hexadecyltrimethylammonium bromide (CTAB)/water, after [50]. L1 : isotropic micellar solution; Ha : hexagonal phase; V1 : bicontinuous cubic phase; La : lamellar phase; C: several heterophasic regions containing crystalline components; NC : nematic phase of rod-like micelles. (b) Cesium pentadeca¯uorooctanoate (CsPFO)/water, after [8].

ture with water, see Figure 14.6(a). For the sake of simplicity, most of the heterophasic regions have been omitted. At low surfactant concentrations and elevated temperatures, the isotropic micellar phase (L1 ) exists. Below a certain temperature there are several heterophasic mixtures of water, ice, and crystalline surfactant (summarized by C). Four liquid crystalline phases can be identi®ed. These are the hexagonal (Ha ), the lamellar (La ), a cubic (V1 ), and a nematic phase (Nc ). Ha consists of rod-like micelles with a twodimensional hexagonal positional order of the long axes of the rods. La stands for stacks of surfactant double layers with intermediate water. The cubic phase V1 is built by two continuous and interwoven networks of surfactant and water channels. Finally, the nematic phase corresponds to the orientational long-range order of rod-like micelles, in analogy to thermotropic nematic states. Only for this latter phase, the formation of a chiral superstructure has been revealed. A second and more peculiar binary system is made up by the anionic surfactant cesium pentadeca¯uorooctanoate (CsPFO), and has been investigated in great detail by Boden et al. [8], see Figure 14.6(b). The corresponding phase diagram contains only three one-phasic regions, namely the isotropic

454

K. Hiltrop

(b) Figure 14.6 (continued )

(L1 ), the lamellar (LD ), and the nematic one (ND ), which in this case consists of disk-like micelles. One outstanding feature of the system is the extremely wide range of stability of the lamellar as well as of the nematic phase. Moreover, the capability to perform phase transitions lamellar/nematic/ isotropic by temperature variation makes the handling of the samples easy. The short, per¯uorinated chains of the surfactant are responsible for the unusual properties of this lyotropic liquid crystal which in some aspects behaves very similar to a thermotropic phase. Both of the brie¯y introduced achiral lyotropics have been used extensively as host phases for the induction of phase chirality by means of chiral dopants.

14.1.4

Chiral Lyotropic Phase Types

Due to the increased degree of freedom with respect to thermotropics the interdependences of molecular properties, external parameters, and phase

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

455

chirality is more complex for lyotropics. Furthermore, the ®eld has not yet been investigated by too many research groups. These arguments may serve as an apology for the fact that the amount of well-understood systems and solved problems is small. Clearly, some of the models and interpretations described in the following are preliminary and hypothetical, but hopefully will encourage us to deal with chiral lyotropics seriously. Up to now the most extensively investigated chiral lyotropics are the chiral nematics (the historical but not appropriate term ``cholesteric'' is avoided in this chapter). The ®rst evidence on this kind of mesophase has been reported by Radley and Saupe in 1978 [9]. Two studies on a lyotropic phase proposed to be analogous to the thermotropic ``Blue Phase'' have been published [10], one has been retracted [11]; if there is any analogy, it is small. Blinov et al. reported on a nonaqueous chiral lamellar phase with ferroelectric properties; experimental evidence for ferroelectricity in lyotropics is di½cult to gain because of high electric conductivity and mostly nonuniform sample orientation. Nevertheless, the existence of piezoelectricity was interpreted as a manifestation of ferroelectricity [12]. Chiral nematics can be achieved in three ways: (i) The micelles are formed by chiral surfactants. Often mixtures of both enantiomers are used, with some enantiomeric excess. (ii) Chiral solvents should also lead to lyotropic phase chirality as has been stated by Osipov [13]. Achievable twists are only small in this case. (iii) Most widely investigated are chiral guest/achiral host systems, because then the variability of accessible structures is largest. Preferable are hydrophobic or amphiphilic dopants. Corresponding to the di¨erent micellar structures and intrinsic order of nematic phases there are three types of chiral nematics: the two uniaxial phases ND (1NL ) and NC , as well as NB , a biaxial phase. In Figure 14.7 the customary models of the micellar arrangement of the uniaxial phases are sketched.

14.1.5

Properties and Methods of Investigation

14.1.5.1

Polarizing Microscopy

All of the lyotropic liquid crystals except the cubic phases exhibit birefringence. Due to their high content of water and the low electronic polarizability of the surfactant molecules the di¨erence, Dn ˆ ne ÿ no , between the extraordinary and ordinary refractive index typically amounts to only 10ÿ3 . Normally merely black and white contrasts can be observed with the polarizing microscope. The sample preparation has to be done in microslides or other completely tight vessels to avoid evaporation of the solvent. Very desirable for the determination of the helical pitch is the formation of ®ngerprint textures, see Figure 14.8.

456

K. Hiltrop

Figure 14.7. Helical arrangement of rod-like and disk-like micelles in the NC and ND chiral nematic phases.

The stripes arise by the in¯uence of the boundary layer for the NL type of chiral nematics, due to the fact that disk-like micelles prefer a ¯at orientation on many, if not all, solid surfaces; adsorption layers of the surfactant form on the solid. Initially hydrophobic surfaces will be covered by a monolayer, the hydrophilic ones by a bilayer. Therefore no spontaneous formation of ®ngerprints occurs with NC -type chiral nematics; instead, Grandjean textures are observed. The periodicity of the ®ngerprints equals half of the pitch, if the sample is prepared correctly. Especially the sample thickness must be large compared to the pitch; otherwise there will be helix unwinding by the action of the solid boundary. Several examples of textures of the rod-like, disk-like, and biaxial chiral nematics are shown and discussed in a paper by Melnik and Saupe [14]. Contact preparations can be used to determine the handedness of the twist if a sample of known handedness is available. The observation of a nonchiral (compensated nematic) region between two twisted parts of the sample is a strong hint to opposite handedness. 14.1.5.2

Magnetic Field Alignment

A magnetic ®eld can be used to enforce ®ngerprint textures also in rod-like nematics, if the sign of the anisometry of the diamagnetic susceptibility of the chiral phase is positive. Then the alignment of the helix axis parallel to

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

457

Figure 14.8. Typical ®ngerprint texture of the ND phase. The periodicity of the stripes corresponds to half of the pitch.

the ®eld is preferred (Dw > 0 for NC phases made up by hydrogen carbon surfactants without aromatic moieties and therefore Dw < 0 for the corresponding NC which results in helix unwinding in strong ®elds). The amount of diamagnetic anisotropy of lyotropics is smaller by a factor of about 100 compared to thermotropics. Therefore a so-called ferro¯uid is added occasionally in order to achieve alignability by small magnetic ®elds (some 100 G) [15]. Ferro¯uids typically consist of suspensions of Fe3 O4 grains (dimensions about 10 nm), coated with oleic acid to improve the stability of the suspension.

458

K. Hiltrop

14.1.5.3

Optical Rotation and Circular Dichroism

As is well known from thermotropics, the handedness of the lyotropic twist can be determined from the sign of the optical rotation. Also the circular dichroism of solubilized dyes has been used for this purpose [16]. 14.1.5.4

Scattering and Di¨raction

X-ray and neutron scattering are suited to determine the size and anisometry of micelles. This gives useful information because there is an in¯uence of these properties on the magnitude of the phase chirality of lyotropics. If the pitch of chiral phases is in the range of a few mm or below, it can be measured by means of laser light di¨raction. Cano preparations have not been used for pitch determination. 14.1.5.5

NMR

The NMR method, especially the deuterium quadrupole splitting can distinguish lyotropic phases, reveal heterophasic regions, determine the order pro®les of the alkyl chains in micelles, and measure the orientational order of molecular moieties in anisotropic surroundings [17], [18], [19].

14.2 14.2.1

Experimental Results and Discussion Systems

A ®rst survey over lyotropic mixtures exhibiting intrinsic as well as induced phase chirality was published in 1989 [20]. In the meantime, a lot of additional systems have been investigated. Chiral surfactants showing intrinsic phase chirality in mixtures with water are listed in Table 14.1. Much more work has been done in the ®eld of phase chirality induced by chiral dopants. Several respective data are collected in Table 14.2. Some corresponding chemical structures are sketched in Figure 14.9.

Table 14.1. Examples of chiral surfactants forming chiral nematic phases.

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

459

Table 14.2. Examples of chiral dopants suitable for the induction of phase chirality in achiral host phases.

460

K. Hiltrop

Figure 14.9. Structures of several chiral dopants investigated hitherto.

14.2.2

Basic Phenomena

The experimentally determined pitches cover the full range down to about 1 mm. Selective re¯ection in the visible range has been tried to achieve but was almost reached only in a few examples [21], [22], mainly for polymeric lyotropic phases [5]. For water-based systems, the twist of lyotropic phases is weaker than for thermotropics by an order of magnitude.

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

Figure 14.9 (continued )

461

462

K. Hiltrop

Figure 14.9 (continued )

14.2.3

Concentration Dependence of the Pitch

Just as for thermotropic cholesterics, the twist pÿ1 increases with increasing enantiomeric excess of a chiral surfactant. The same holds for the dopant concentration. The capacity of solubilization within the micelles depends on the physico±chemical nature of the dopant and host phase; especially the size and amphiphilicity of the dopant are essential. The course of the twist pÿ1 versus dopant concentration x is linear for small x; the initial slope ……q pÿ1 =qx†x!0 † is de®ned as helical twisting power (HTP) as for thermotropics. Figures 14.10 and 14.11 show typical experimental data for a system consisting of chiral surfactants and for a guest/host system, respectively. For the chemical structures of the dopants of Figure 14.11, see Figure 14.9. The course of pÿ1 versus x becomes nonlinear at elevated dopant concentrations; mostly the absolute value of the slope deceases but some exceptions are known [23]. For a few cases a maximum twist has been found, especially when a lamellar phase is adjacent to the chiral nematic one. Then pretransitions obviously occur close to the lamellar state, as has been shown already in the basic paper by Radley and Saupe [9], [24]. So far, lyotropic chiral nematics seem to behave quite similar to their thermotropic counterparts.

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

463

Figure 14.10. Twist pÿ1 versus concentration in a mixture of two chiral surfactants, after [25].

However, there is an essential di¨erence to thermotropic cholesterics: the size and shape of the building blocks of the phase, which are the micelles, are subjected to substantial variations, see Section 14.2.6. At ®rst glance, the helical twisting power is a property of the dopant itself; but it is also strongly in¯uenced by the achiral host phase. The latter e¨ect is due to several reasons. Among these are the elasticity of the host, correlated to the size and shape of the micelles, but also to solubilities and locations of the dopants with respect to the aggregates, see Section 14.2.7.

464

K. Hiltrop

Figure 14.11. Twist versus concentration x of some chiral dopants in an achiral host phase made up by hexadecyldimethylethylammonium bromide/decanol/water, after [23]; for the dopant structures see Figure 14.9.

14.2.4

Handedness

The handedness of the twist of a chiral lyotropic phase depends in a complex manner on various properties of the chiral and the achiral matter of the mixture. Very little is known about the physical origin of the observed dependences. Up to now it is not possible to predict the handedness of an arbitrary system, not even in every case within the homologeous series of surfactants or dopants. For sure, inversion of the absolute con®guration of a chiral dopant leads to helix inversion of the phase chirality. Two isomeric chiral surfactants with alanine headgroups of the same absolute con®gurations have been investigated in mixtures with similar compositions, but with various relative amounts of the two isomers, see Figure 14.10. The handedness of the phase chirality turned out to be di¨erent for these isomers; this was explained by the opposite relative con®gurations of the chiral center with respect to an alkyl chain [25], [26]. Radley et al. [27] proved an inversion of the handedness induced by variation of the host phase. All of the used surfactants were alkylmethylammonium halides with a di¨erent number of methyl moieties and chain lengths. A derivative of alanine induced a positive twist only in solutions of tetrade-

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

465

cyltrimethylammonium bromide, but negative ones in dodecyldimethylammonium bromide, dodecylmethylammonium chloride, and decylammonium bromide. It was proposed that this behavior is due to the existence of two di¨erent chiral conformations of the dopant, which are thought to possess helical twisting powers of di¨erent magnitude and sign. The relative amount of two rotamers (cis- and trans-) were determined by means of 13 C-NMR. A nonlinear but monotonous dependence of the helical twisting power on the mole fraction of one of the two rotamers was found. In a former investigation, an inversion of the helix sense with dopant concentration has been observed which is unusual. The phenomenon was explained again by a change of the cis/trans ratio of the surfactant (the chloride of the decyl ester of L-proline). NMR gave at least qualitative evidence for the existence of the two rotamers [28]. An inversion of the twist sense has been found with a-hydroxy carbon acids and some corresponding potassium salts [23]. For all of the investigated couples the respective opposite handedness was proven by a twist compensation seen in contact preparations. The result is unexpected because the molecular chirality of the acid and the salt are nearly identical. It was supposed that the presumably different locations with respect to the micelles due to the ionic character of the salt and/or a dimerization of the acid could be the reason for the observed behavior.

14.2.5

Temperature-Dependence

The twist of thermotropic as well as of lyotropic liquid crystals depends on temperature. For thermotropic cholesteric phases, the pitch decreases with increasing temperature in most cases. For lyotropics, both negative and positive temperature coe½cients occur frequently, sometimes with complex courses, depending on the chemical constitution and the composition of the mixture [29], see Figure 14.12(a). Approaching a phase transition chiral nematic/lamellar forces the pitch to increase strongly, see Figure 14.12(b). This behavior is obviously due to pretransitional e¨ects and is correlated either with micellar growth or, from a macroscopic point of view, with an increase in the twist elastic constant k2 . Radley and Tracey [30] discussed the decreasing twist with increasing temperature for the chiral surfactant potassium N-dodecanoyl-l-alaninate with respect to two processes originally proposed for thermotropics [31], [32]. These are increased intermolecular distances on the one hand, and an increased local twist on the other hand, as a consequence of enhanced thermal motion. The sign of the temperature coe½cient of the twist is opposite for the two e¨ects. If any of these mechanisms ®ts to lyotropics remains speculative, because the question for the ``mechanisms'' of the transfer of chirality within these systems is not answered. In addition to the above-mentioned discussion, a temperature-dependence of the population of di¨erent conformational states of the chiral dopants could a¨ect the twist. Furthermore, a more

466

K. Hiltrop

(a)

(b) Figure 14.12. (a) Temperature-dependence of the pitch for three di¨erent a-hydroxy carbon acids (see Figure 14.9) in identical host phases (CDEABr/decanol/water), after [51]. (b) Temperature-dependence of the pitch close to the chiral nematic/lamellar phase transition; curve parameter: host phase composition wt/wt; chiral dopant: tomatine.

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

467

complex in¯uence of temperature must be suspected, e.g., by variation of the size of the micelles, the solubilization of the dopants, etc.

14.2.6

Size and Shape of the Micelles

Concentration of all components of a lyotropic system (especially the solubilization of dopants [33]), temperature, pH, etc., can a¨ect the size and shape of the aggregates. By means of X-ray scattering one can determine approximately the shape anisotropy of micellar aggregates [15]. This has been done for the system potassium laurate/decanol/water/brucine sulfate which forms disk-like micelles in a certain composition/temperature range. Changing the relative amounts of surfactant/cosurfactant, X-ray data revealed that the plane-to-plane distance dy was constant for all samples; the in-plane periodicity of the disk-like micelles changed with the decanol content of the mixture: more decanol lead to larger dx . On the premises of an equal thickness of the water layer covering each micelle, larger dx means micelles with larger diameter. The authors found a correlation of a greater shape anisometry of the micelles to a smaller pitch. Yet the accessible range of anisotropies was only 1.2 to 1.5, i.e., the micelles have been rather small in this case and the observed behavior is plausible. A pitch decrease with increasing salt concentration and with decreasing pH has been found by Labes et al. [34]. The addition of salt to a solution of ionic surfactants reduces the Debye length of the ions; therewith the headgroups become e¨ectively smaller and the micelles grow. Larger micelles will ®nally decrease the twist of the director ®eld; the extreme case is the lamellar state which does not exhibit phase chirality. In the case of 2-octanol, both the d- as well as the l-con®gurated enantiomers induced the same decrease of twist in a chiral host system [35]; the explanation is presumably that the HTP of 2-octanol is small and it acts mainly as a cosurfactant inreasing the size of the aggregates. Surely some optimum value of the size of the micelles must be expected with respect to a maximum HTP. In summary, the addition of a chiral dopant not only induces phase chirality but also modi®es several achiral properties of the phase. A simple possibility to avoid uncontrollable consequences by changes of the composition of a chiral lyotropic mixture is the simultaneous use of both enantiomers of a chiral surfactant or dopant and to vary only the enantiomeric excess. For racemates, compensated mixtures are formed [36], as known from thermotropics.

14.2.7

Structure/Property Relations of Dopants and Host Phases

From the twisting e½ciency found for tartaric acid, brucine sulfate, and cholesterol, Saupe and Radley concluded that the twisting power of a chiral dopant is large only if it is solubilized within the micelles. The HTP was

468

K. Hiltrop

supposed to be directly proportional to the distribution coe½cient [9]. In qualitative agreement with this statement, Gottarelli et al. [16] reported that readily water-soluble dopants like sacharose and glucose, as well as a charged metal comlex (L-Co(ethylendiamine)3 Cl3 ), do not exhibit large HTPs. Interestingly, an uncharged similar complex (L-Rh(pentandionate)3 ), with some solubility in organic solvents, which will therefore be solubilized within the micelles, e¨ectively twisted a nematic host phase. Before going into more detailed discussion of the structural properties, a survey over some parameters, which are essential for the induction of phase chirality in lyotropic liquid crystals, is given in Table 14.3. In the literature there are only a few studies dealing systematically with the in¯uence of certain properties of the chiral surfactants or dopants on the phase chirality. Using a chromonic host phase (disodium chromoglycate, for the structure see Figure 14.13), Labes et al. [37] derived some rules by investigating many di¨erent amino acids. It turned out that long and bulky side-chains within the dopant molecule result in small HTPs; on the other hand, a strong hydrophilic side-chain brings about an increased HTP [22]. The main point behind these rules is that long and bulky side-chains experience a strong sterical hindrance and therefore would lead to increased intermolecular and interaggregate distances; large pitches were the consequence. On the other hand, hydrophilic groups make up hydrogen bonds with water or surfactant molecules and thus behave contrarily to the bulky side-chains. In principle, it is not certain if these rules, which have been found for one chromonic host phase, can be generalized to other chromonic and, even more, to micellar systems also. A hollow column of about 1.6 nm diameter is proposed as the structure unit of this chromonic phase. The aggregates of this kind of lyotropics are distinctly di¨erent from micelles. For micellar systems, a lot of data has been measured for di¨erent homologues of a-hydroxy carbon acids and salts in various host phases [23], [29]. The used dopants are advantegeous because they possess only one chiral center; one substituent of the chiral carbon could be varied widely by simple chemistry, see Figure 14.9. An in¯uence of the molecular moieties bound to the chiral center became obvious. A large electronic polarizability and/or bulkiness of this moiety favors a large HTP. Similar results had been found for thermotropic induced cholesteric mixtures [38]. The in¯uence of the host phase was checked for some homologues of the anionic alkylammonium halides as well as the cesium and ammonium salts of per¯uorinated carbon acids which are cationic surfactants. The results can be summarized by two statements: (i) the longer the alkyl chain of the surfactant, the smaller the twist; and (ii) the bulkier the head group of the surfactant, the smaller the twist, see Figure 14.14. Extremely large twists have been detected with some sugar/steroid compounds like digitoxine, solanine, and tomatine. Figure 14.15 shows that a nonzero twist can be measured already at very low dopant concentrations. Assuming an aggregation number of 200 per micelle, a dopant concentration

14. Phase Chirality of Micellar Lyotropic Liquid Crystals Table 14.3. Parameters of the induction of phase chirality.

469

470

K. Hiltrop

Figure 14.13. Structure of disodium chromoglycate which forms a chromonic phase in mixtures with water.

of 0.01 mol% corresponds to an average number of 1 dopant molecule per 50 micelles. The large number of chiral carbons presumably is not the reason for the large HTP [48], [39]. Instead, compensation e¨ects of di¨erent chiral centers must be expected; evidence for this behavior has been observed comparing the chiral surfactant potassium dodecanoyl threoninate (two chiral carbons) with the homologeous alaninate (one chiral carbon) [17]. A look at Figure 14.9 and Table 14.2 gives a hint of the possible importance of hydrogen bonding. Up to now, no systematic study of this point is known. 14.2.7.1

Location of the Dopant within the Aggregate

Vanin et al. [40] used dopants with di¨erent locations of mesophase environments; they found that only when the dopant is solubilized mainly in the bulk water its structure is important with respect to the twist sense of di¨erent host phases. The use of strongly water soluble sugars exhibited a very low twisting e½ciency. In their studies hydrophobic dopants lead to the same handedness in all investigated hosts (decylammonium chloride, potassium alkanoates, sodium, and cesium alkylsulfates). To discuss their results they proposed three di¨erent situations: (1) the hydrophilic dopant remains completely outside the aggregates, and its own structure (constitution, con®guration, conformation) determines the twist in magnitude and sign; (2) the hydrophobic dopant is solubilized completely in the hydrocarbon core of the micelle and should induce the same twist in all di¨erent host phases; (3) the dopant interacts with the electric double layer of the aggregate and either the hydrophilic ``inducer'' anchors at the micellar surface or the hydrophobic ``inducer'' is present in the micellar core, but an achiral ionic group anchors in the double layer. In both cases, the twist will be in¯uenced by special dopant/micelle properties. The interaction of the electronic double layer with the chiral dopant should play an important role.

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

471

Figure 14.14. In¯uence of the host phase on the helical twisting power of chiral dopants, after [29]; CnCm: alkylammonium bromides with di¨erent chain lengths; A±Cn (resp. Cs±Cn): ammonium (resp. cesium) per¯uoroalkanoates.

472

K. Hiltrop

Figure 14.15. Twist versus dopant concentration for CsPFO/water doped with tomatine, after [48].

A problem with the generally assumed solubilization of the dopants within the host micelles becomes obvious by regarding tomatine and similar compounds in CsPFO/water: this dopant cannot be solubilized within standard micelles of about 200 monomers of surfactant for sterical reasons, see Figure 14.16. Clearly, a grossly modi®ed aggregate must be formed. 14.2.7.2

Orientation of the Chiral Dopant

The headgroup alignment of potassium N-dodecanoyl-l-alaninate has been studied by NMR [18]. The order parameter of the alaninate moiety turned out to be inversely proportional to the twist …p z S† which is an unexpected result and ®nally was interpreted by the assumption that the rotational axis of the head group approached the magic angle orientation in concert with the layer twist. A further question was if the change of headgroup orientational order is accompanied by some variation within the aliphatic chain of the surfactant. Indeed, the respective deuterium quadrupole splittings exhibited a related change of the preferred conformation of the amphiphile. The order parameter for all positions of the aliphatic chain decreased by a

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

473

Figure 14.16. Sterical models of the surfactants hexadecyldimethylethylammonium bromide (CDEABr) and cesium pentadeca¯uorooctanoate (CsPFO) for comparison with the dopant tomatine.

factor 3 on increasing the amphiphile concentration; a similar behavior also occurred for the head group, whereas the counterion signal remained constant. A correlation between twist, headgroup alignment, and conformation of the aliphatic chain seems to be conclusive from these data. On the other hand, the variation of the pitch caused by temperature changes was not correlated with similar selective changes in headgroup orientation which again is unexpected. An extensive amphiphilic character of a chiral dopant is believed to favor an e¨ective induction of phase chirality. Amphiphilic dopants exhibit an increased orientational order within their host aggregates, as compared to merely hydrophilic or hydrophobic molecules. The observed phenomena as well as the hypothesis ®t qualitatively to the model of an intra-micellar double-twist [23], see Section 14.3.

474

K. Hiltrop

14.2.8

Miscellaneous

In a lyotropic system a case of real ferroelectricity of liquid crystals has been detected. The origin of the ferroelectricity in solid crystals is a dipole±dipole interaction. Poly-l-glutamate forms an a-helix which is a rather rigid rod due to intramolecular hydrogen bonds. The macromolecule exhibits a large dipole moment along the molecular axis [41]. Cellulose derivatives have been mixed with achiral nematic host phases made up by anionic or cationic surfactants. Only for the noncharged cellulose and only for disk-like micelles a strong HTP could be observed. The HTP of the polymer is much higher than that of low molecular weight dopants, presumably a kind of necklace of individual micelles forms, see Figure 14.17 [42].

14.3

Mechanisms

Together with the report on the discovery of lyotropic chiral nematics, the very ®rst proposals for mechanisms of chiral interaction were published by Radley and Saupe [9]. The authors discussed two models: (i) the micelles adopt a chiral shape and thereby give rise to phase chirality via sterical interaction between adjacent micelles; and (ii) a chiral dispersion force acts directly between chiral molecules located in di¨erent micelles.

Figure 14.17. Chiral necklace of micelles induced by a chiral polymer chain, after [42].

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

475

Figure 14.18. Proposals for the mechanisms of chirality transfer between micelles, (a) after [43].

Both models are sketched in Figure 14.18. A rod-like micelle can be imagined to have a shape like a corkskrew, for a disk-like micelle it is not as obvious how it can be twisted without disruption. Strayley extended the ideas of Onsager to chiral particles (not necessarily micelles) [43]. Onsager had shown that a disordered arrangement of anisometric particles costs translational entropy with increasing concentration of the particles. The increase of translational entropy can be avoided by the introduction of orientational order. Particles having a chiral shape consume less space by packing in a twisted manner predetermined by their shape, see Figure 14.18(a). Plots of the dependence of the twist on the inverse temperature would reveal the entropic contribution to the chiral interaction which has also been considered by Osipov [13]. In order to describe chiral dispersion forces, macromolecules as well as micelles were modeled as anisometric bodies ®lled with a dielectric medium. The interaction of such particles is then determined through ¯uctuations of the electromagnetic ®eld which can be in¯uenced by molecular chirality. Within this frame the action of a chiral solvent can be understood, too. Reeves et al. argued that a dispersion type of interaction between single chiral molecules in adjacent micelles is rather unlikely to be essential for the phase chirality [35]. Dispersion interactions between colloidal particles, as well as between single molecules, decrease with a strong power of the interparticle distance d (by d ÿ4 to d ÿ6 ). In thermotropic cholesteric or chiral smectic phases the interparticle distance is of the order of 100 pm which is small compared to the typical values of 1 to 10 nm between micelles in lyotropic chiral nematic phases.

476

K. Hiltrop

Some disk-like nematics exhibit a structure which is locally similar to that of a lamellar phase; X-ray low-angle scattering data contain peaks of second and even higher order; the local lamellar structure may extend to about ®ve layer periods [44]. Levelut et al. indicated that even in the chiral nematic state the pseudo-lamellar microscopic structure is maintained [45]. On the other hand, they detected a di¨erence in the re¯exes of the NC and the unwound NC phase, which can be explained either by a distortion of the local lamellar order by the in¯uence of the chiral dopant or by a distortion of the shape of the micelle due to the bulkiness of the dopant (brucine sulfate, see Figure 14.9). Figueiredo Neto et al. suggest that large hydrophilic dopants like brucine sulfate might be located between adjacent micelles decreasing the intermicellar interactions and prevent the orientational ¯uctuations around the axis perpendicular to the amphiphilic bilayer [46]. In that way, large chiral dopants could serve as a kind of string for the pearls of a necklace as has been found for a polymeric dopant [42]. Vanin et al. studied order pro®les of the perdeuterated laurate chain in chiral and achiral phases by means of deuterium NMR. The quadrupole splittings measured for the sequence of CD2 moieties are all quite similar for the nematic and chiral nematic phase; the conclusion was that chiral distortions of the micellar shape must be very small. Their proposal for the mechanism behind the phase chirality is analogous to the one for polypeptides: an asymmetric point charge distribution which in this case was speculated to be created by the chiral dopant on the micellar surface [19]. Radley and Tracey prepared samples from a racemic detergent and observed high ¯uidity, whereas those consisting of some enantiomeric excess showed a signi®cantly increased viscosity [30]. But this is no proof for the association of micelles to necklaces. A theoretical approach to the observed temperature e¨ects is achieved from the macroscopic aspects given by Osipov [13]; he stated that the two proposed chiral interactions which are possibly the basic mechanisms for lyotropic phase chirality have di¨erent signs. Their action leads to opposite twist senses of the macroscopic helix. In principle, temperature-induced helix inversions could be explained. In summary it must be stated that the development of models of the mechanisms relevant to phase chirality in lyotropic liquid crystals has not yet succeeded in giving a conclusive explanation. Referring to the interplay of physical properties and twisting e½ciency as sketched in Table 14.3, the following two steps have been taken.

14.3.1

Intramicellar Chirality

Several experimentally observed similarities between thermotropic and lyotropic induced phase chirality suggested a structural relationship. This was the origin of the hypothesis of an intramicellar chirality in disk-like micelles. At ®rst, a disk-like micelle is simpli®ed to a piece of a double-layer of ®nite

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

477

Figure 14.19. Model of an intramicellar chirality, after [48].

size which is made up by achiral surfactants. Due to their amphiphilicity the surfactants exhibit a local orientational order. By addition of a chiral dopant, the locally ordered assembly of long molecules was suspected to adopt a chiral superstructure, similar to the local order of induced cholesteric phases. Because of the ®nite size of the micelle the chiral order could develop without frustration in the plane of the disk; then a double-twist cylinder is formed [23], see Figure 14.19. Anyway, the experimental veri®cation (or falsi®ca-

478

K. Hiltrop

tion) has not yet been performed. Moreover, the intermicellar chirality transfer is still an open problem.

14.3.2

Quanti®ed Chirality

Chirality is not only a quality. Several possibilities to realize a measure de®ning the quantity of chirality were reviewed by Mislow et al. [47]. The authors proposed a special measure, the so-called ``Hausdor¨ disctance,'' to be well suited to describe the chirality of a topological object. Simply speaking, the Hausdor¨ disctance speci®es the dissimilarity of an arbitrary object with respect to its mirror image. The respective formalism has been adopted to the case of chiral dopant molecules; these were represented by the coordinates of their atoms as calculated for the lowest-energy conformation by means of a molecular modeling computer software. However, the resulting chirality parameter of several investigated dopants did not correlate with the corresponding HTP values. It was suggested that a strong chirality of a dopant might be a necessary but not su½cient property to achieve a large twisting power, unless the transfer of chirality between dopant and host phase is actually e¨ective. The importance of a ``similarity'' between the interacting moieties was demonstrated [48].

References [1] A. Sonin, Uspekhi Fiz. Nauk. 153, 273 (1987). [2] R. Meyer, in: Dynamics and Patterns in Complex Fluids (edited by A. Onuki and K. Kowasaki), Springer Proc. Phys., p. 62, (1990). [3] N. Usol'tseva, K. Praefcke, D. Singer, and B. GuÈndogan, Liq. Cryst. 16, 617 (1994). [4] T. Samulski and E. Samulski, J. Chem. Phys. 67, 824 (1977). [5] P. Zugenmaier and P. Haurand, Carbohydr. Res. 160, 369 (1987). [6] C. Tanford, The Hydrophobic E¨ect, Wiley, New York, 1973. [7] J. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1985. [8] N. Boden, S. Corne, and K. Jolley, J. Phys. Chem. 91, 4092 (1987). [9] K. Radley and A. Saupe, Mol. Phys. 35, 1405 (1978). [10] K. Radley, Liq. Cryst. 18, 151 (1995). [11] H. Lee and M. Labes, Mol. Cryst. Liq. Cryst. 82, 355 (1983). M. Kuzma, H. Lee, and M. Labes, Mol. Cryst. Liq. Cryst. 92, 81 (1983). [12] L. Blinov, S. Davidyan, A. Petrov, A. Todorov, and S. Yablonskii, JETP Lett. 48, 285 (1988). [13] M. Osipov, Il Nuovo Cimento 10, 1249 (1988). [14] G. Melnik and A. Saupe, Mol. Cryst. Liq. Cryst. 145, 95 (1987). [15] M. Valente Lopes and A. Figueiredo Neto, Phys. Rev. A 38, 1101 (1988). [16] G. Spada, G. Gottarelli, B. Samori, C. Bustamente, and K.S. Wells, Liq. Cryst. 3, 101 (1988).

14. Phase Chirality of Micellar Lyotropic Liquid Crystals

479

[17] A. Tracey and K. Radley, Langmuir 6, 1221 (1990). [18] A. Tracey and X. Zhang, J. Phys. Chem. 96, 3889 (1992). [19] M. Alcantara, M. de Melo, V. Paoli, and J. Vanin, Mol. Cryst. Liq. Cryst. 95, 299 (1983). [20] M. Boidart, A. Hochapfel, and P. Peretti, Mol. Cryst. Liq. Cryst. 172, 147 (1989). [21] A. Tracey and K. Radley, J. Phys. Chem. 88, 6044 (1984). [22] H. Lee and M. Labes, Mol. Cryst. Liq. Cryst. 108, 125 (1984). [23] J. Partyka and K. Hiltrop, Liq. Cryst. 20, 611 (1996). [24] M. Acimis, E. Dorr, and H. Kuball, Liq. Cryst. 17, 299 (1994). [25] K. Radley and H. Cattey, Mol. Cryst. Liq. Cryst. 250, 167 (1994). [26] K. Radley and H. Cattey, Mol. Cryst. Liq. Cryst. 226, 195 (1993). [27] K. Radley, N. McLay, and K. Gicquel, J. Phys. Chem. 101, 7404 (1997). [28] K. Radley and N. McLay, J. Phys. Chem. 98, 3071 (1994). [29] M. Pape and K. Hiltrop, Mol. Cryst. Liq. Cryst. 307, 155 (1997). [30] K. Radley and A. Tracey, Can. J. Chem. 63, 95 (1985). [31] P. Keating, Mol. Cryst. Liq. Cryst. 8, 815 (1969). [32] H. Kimura, M. Hosini, and H. Nakano, J. Phys. C3, 40, 174 (1979). [33] H. DoÈr¯er and C. Swaboda, Tenside Surf. Det. 35, 126 (1998). [34] R. Goozner and M. Labes, Mol. Cryst. Liq. Cryst. 116, 309 (1985). [35] P. Covello, B. Forrest, M. Marcondes Helene, L. Reeves, and M. Vist, J. Phys. Chem. 87, 176 (1983). [36] M. Acimis and L. Reeves, Can. J. Chem. 58, 1533 (1980). [37] D. Goldfarb, M. Moseley, M. Labes, and Z. Luz, Mol. Cryst. Liq. Cryst. 89, 119 (1982). [38] H. Finkelmann and H. Stegemeyer, Ber. Bunsenges. Phys. Chem. 78, 870 (1974). [39] H. DoÈr¯er and C. Swaboda, Tenside Surf. Det. 35, 18 (1998) . [40] T. Henriques do Aido, M. Alcantara, O. Felippe, A. Galvao Pereira, and J. Vanin, Mol. Cryst. Liq. Cryst. 195, 45 (1991). [41] B. Park, Y. Kinoshita, H. Takezoe, and J. Watanabe, Jpn. J. Appl. Phys. 37, L136 (1998). [42] Y. Yarovoy and M. Labes, J. Am. Chem. Soc. 119, 12109 (1997). [43] J. Strayley, Phys. Rev. A 14, 1835 (1976). [44] M. Holmes, M. Leaver, and A. Smith, Langmuir 11, 356 (1995). [45] A. Figueiredo Neto, A. Levelut, Y. Galerne, and L. Liebert, J. Phys. France 49, 1301 (1988). [46] M. Marcondes Helene and A. Figueiredo Neto, Mol. Cryst. Liq. Cryst. 162B, 127 (1988). [47] A. Buda and K. Mislow, J. Am. Chem. Soc. 114, 6006 (1992). [48] E. Figgemeier and K. Hiltrop, Liq. Cryst. 26 (1999). [49] G. Bartusch, H. DoÈr¯er, and H. Ho¨mann, Progr. Colloid Polymer Sci. 89, 307 (1992). [50] T. Wol¨ and B. Klaussner, Adv. Colloid Interface Sci. 59, 31 (1995). [51] E. Figgemeier, M. Pape, and K. Hiltrop, unpublished results.

480

14.4

K. Hiltrop

Abbreviations

CDEABr: Hexadecyldimethylethylammonium bromide. CsDS: Cesium decylsulfate. CsPFO: Cesium pentadeca¯uorooctanoate. DACl: Decylammonium chloride. DSCG: Disodiumchromoglycate. KL: Potassium laurate. SdS: Sodium decylsulfate. SDS: Sodium dodecylsulfate. TDMAO: Tetradecyldimethylamineoxide.

15

Traveling Phase Boundaries with the Broken Symmetries of Life P.E. Cladis

To honor Professor Heppke's long interest in, and many contributions to, the study of chiral liquid crystalline materials, phase diagrams, and life, this chapter is an overview of our work on the traveling cholesteric (N*)isotropic phase boundary [1], [2].

15.1

Introduction

Living systems are the most important and most complex of nonequilibrium liquid crystalline systems. We check an experimental minimal model for macroscopic implications of this statement. To do this, we consider ``living systems'' in terms of three broken symmetries: (1) broken continuous spatial symmetryÐliquid crystalline; (2) broken mirror symmetryÐchiral; and (3) broken time reversal symmetryÐnonequilibrium. We call these three elements the Broken Symmetries of Life. The surprising result is that a model system with just these three elements, also ``knows time'' [2].

15.1.1

Broken Continuous Spatial Symmetry

The ®rst important characteristic is that living systems are liquid crystalline. Liquid crystals are orientationally ordered molecular liquids. When a molecular liquid is in the isotropic liquid state (which models the primordial soup), it has neither translational nor orientational order. At the phase transition to the nematic state, the most basic liquid crystal transition, the system selects a special direction for long-range orientational order. This direction is called the director and denoted by a unit vector, n. By selecting a special direction, the nematic breaks the continuous rotational symmetry of the isotropic liquid. A phase boundary intervenes between the nematic and the isotropic liquid when they coexist. 481

482

P.E. Cladis

15.1.2

Broken Mirror Symmetry

The second living system feature we select is that they are chiral. We take this to mean broken mirror symmetry. In liquid crystals, chirality is expressed as a macroscopic helix with a pitch, p0 . In cholesteric liquid crystals, Ê < p < y, and q , its wavevector (wavenumber q0 ˆ 2p=p ), is per1500 A 0 0 0 pendicular to n, the direction of orientational order. The existence of this intrinsic equilibrium length has several implications of which we mention two particularly insightful ones [1], [2]. The ®rst is that orientational di¨usion in cholesteric liquid crystals is characterized by a di¨usion constant, D0 1 K2 =g1 , where K2 is the twist elastic constant and g1 the rotational viscous coe½cient. In a system with an equilibrium length scale, q0 , a characteristic frequency oel ˆ D0 q02 can be de®ned. By comparing the magnitude of this frequency with experimental observations, we can assess the importance of the various di¨usion processes at work in a given situation. If the observed o is such that o=oel @ O…1†, this tells us that the most important dissipative process involved at the traveling cholesteric±isotropic phase boundary is orientational di¨usion, as indeed it turns out to be for patterns closer to equilibrium [1]. The second is that a mirror transformation such as z ! ÿz results in q0 ! ÿq0 : in a mirror, a right hand is a left hand. As pointed out by Brand and Pleiner [3], the handedness of q0 allows terms in the dynamic equations coupling q0 to gradients of a scalar such as `T, a temperature gradient, or `cy , gradients in concentration. This coupling involving impurity di¨usion with a di¨usion constant, DI , dominates at the ¯at phase winding interface observed further from equilibrium where the impurity gradient is very steep [2].

15.1.3

Broken Time Reversal Symmetry

A third feature of living systems is that they are a consequence of many diffusive processes that can only be sustained by continuous energy input. Living systems are far from equilibrium where no energy input is required. They are nonequilibrium systems characterized by broken time reversal symmetry. How do we model these three broken symmetries of life before being carried o¨ to be burned at the stake?

15.2

A Minimal Model System

Arguments for the minimal model system being the traveling cholesteric liquid crystal±isotropic liquid phase boundary are given in this section. The sample is prepared so that the length scale imposed on it by the nonequilibrium driving forces is comparable to the cholesteric liquid crystal's equilibrium length, its pitch. In a pattern formation context, ``knows time''

15. Traveling Phase Boundaries with Broken Symmetries of Life

483

means that a frequency is associated with all the patterns of this traveling interface. With the same nonequilibrium driving forces, its nonchiral analogue, the traveling nematic±isotropic liquid phase boundary, does not ``know time.''

15.2.1

Liquid Crystalline

The ®rst Broken Symmetry of Life, we identi®ed with a liquid crystal± isotropic liquid transition. Something (i.e., the liquid crystal order parameter) emerges from nothing, the primordial soup.

15.2.2

Chiral

To model the second element of Life's Broken Symmetries, we chose cholesteric liquid crystals (N*) as they are chiral liquid crystals. We made C15±8CB cholesteric±nematic mixtures where the concentration, cy , of C15 in 8CB lowered the cholesteric±isotropic transition temperature by TChI … C† ˆ 40:36 ÿ 0:22cy (%). The pitch decreased with cy as q0 …mmÿ1 † ˆ 0:02cy (%): the more C15, the lower the transition temperature and the tighter the pitch and vice versa. The sample cell was the usual two parallel glass plates assembly with separation d A p0 A 38 mm. The glass plates were prepared so that n was uniform and in the plane of the glass.

15.2.3

Nonequilibrium

While pattern formation, epitomized by beautiful snow¯akes, was ®rst liberated from theology by Kepler in 1611, it is only relatively recently some progress has been made in its scienti®c investigation [4]. Pattern formation refers to spatial structures, usually characterized by a band of wavelengths, that replace a uniform state above a critical distance from equilibrium. Patterns are nonequilibrium structures resulting from several dissipative processes characterized by di¨usion constants. They are quanti®ed by their wavevector, q, and their frequency, o. To put the N*±iso transition line into a nonequilibrium situation, we force it to move at a velocity, v, through a temperature gradient Gkv. The temperatures at the hot and cold contacts are chosen so that TChI is in the middle of the ®eld of view of a polarizing microscope (Figure 15.1). We force v on the interface by displacing the sample at speed, ÿv, toward the cold contact. As the transition temperature is ®xed in the lab frame, displacing the sample toward colder temperatures forces the N*±iso phase boundary to move toward the hot contact to maintain TChI . This is a well-known technique called Directional Solidi®cation. It was invented by Jackson and Hunt to advance industrial crystal growth processes [5]. Improvements in our setup, compared to their original one, include several computer controled stepping motors and image processing software.

484

P.E. Cladis

Figure 15.1. The sample is loaded with n at the surfaces C nkG, the temperature gradient. Left: Experimental setup for directional solidi®cation. Gkv, the velocity of the N*±iso phase boundary as the sample is displaced from the hot contact to the cold one at speed, …ÿv†. Center: The ¯at interface when v < vc ˆ 19 mm/s. As z ˆ 0 at the interface and T …K† ˆ Gz, the picture dimensions are 250 mm  0:25 K. Right: 510 mm  0:3 K picture of the modulated interface. Its wavelength, l ˆ 104 mm when v ˆ 25 mm=s > vc . Those parts of the interface toward higher temperatures (tips) are less rich in C15 therefore have a larger pitch than those at lower temperatures (grooves). The two headed arrows highlight the transition line. Note the increase in the N*±iso meniscus in the moving sample.

15.3 15.3.1

Directional Solidi®cation in Liquid Crystals General Remarks

A phase boundary forced to travel by mechanical displacement at speed, ÿv, in a temperature gradient, G, is a model pattern forming system (Figure 15.1). It has the advantage that both G and v can be well controlled. Because liquid crystals are a strategic component of the semiconductor industry, they are well characterized. Consequently, reliable information can be obtained from the various patterns exhibited by traveling liquid crystal phase boundaries. Another important strength of cholesteric and nematic liquid crystals in this context is that they are three-dimensional liquids. As a result, transitions from equilibrium to turbulence can gracefully evolve, so can be followed in the lab frame. All length and time scales do not suddenly appear at one transition, or bifurcation, as they do at, e.g., the traveling TGBA ±smectic A phase boundary [6].

15. Traveling Phase Boundaries with Broken Symmetries of Life

15.3.2

485

Cholesteric Liquid Crystals

What are the processes at work to make a pattern in directional solidi®cation? As the interface advances, it expels impurities into the isotropic liquid. The impurity excess at the interface follows linear di¨usion to exponentially decay from a maximum at the interface …cI † to cy far from the interface: c…z† ˆ :cy ‡ Dc  exp…ÿz=lD †. Dc ˆ cI ÿ cy . The characteristic length for this decay is the dynamic di¨usion length, lD ˆ DI =v @ 2 mm when v ˆ 20 mm/s. DI is the impurity di¨usion constant, DI ˆ 2  10ÿ7 cm 2 /s [7]. Here the impurity is C15, the chiral agent. In liquid crystals, setting up a steady state concentration gradient in front of the moving interface is relatively fast with a characteristic time, tD ˆ DI =v 2 @ 0:2 s when v @ vc ˆ 19 mm/s. Impurity di¨usion is the only destabilizing nonequilibrium force in this system. Above a critical speed, vc , the destabilizing force of the impurity gradient, overcomes the stabilizing in¯uences of the temperature gradient and elastic forces, so that a pattern characterized by both a q < q0 and frequency, o=2p, form at the interface (Figure 15.1, right).

15.3.3

Constitutional Supercooling

The classic argument for why a ¯at traveling phase boundary becomes unstable to a pattern in directional solidi®cation is called constitutional supercooling [9]. The argument is easily understood when one keeps in mind that a temperature gradient maps temperature onto space: T…K† ˆ Gz (Figure 15.1). A pertubation of the interface into the hotter region puts it in contact with isotropic liquid at temperature T…z† and concentration c…z†. If the concentration gradient does not fall steeper than the equilibrium cy ±TChI slope, this isotropic liquid is warmer than the interface so the bump melts stabilizing the ¯at interface. As v increases, the dynamic di¨usion length, lD , decreases so that at vc the concentration gradient is steeper than the equilibrium cy ±TChI slope. In this case, a perturbation now puts the interface in contact with isotropic liquid that is purer than it is supposed to be, i.e., the isotropic liquid is at a lower temperature than dictated by cy ±TChI . It is locally supercooled because its impurity level is too small for its temperature. Thus the name constitutional supercooling. This is an unstable situation so the bump on the interface grows into the isotropic liquid nucleating a pattern called a cellular pattern (Figure 15.1, right).

15.3.4

Wavelength Selection

While constitutional supercooling explains why one bump grows, it does not account for a line of bumps with wavenumber q. The outstanding question of pattern formation is still: Where does q come from? Nevertheless, at the

486

P.E. Cladis

traveling nematic±isotropic phase boundary and the traveling N*±iso phase boundary, the cellular pattern has a length scaleÐnot a bandÐthat is a result of nonequilibrium forces acting on the system.

15.4

The Traveling Cholesteric±Isotropic Interface ``Knows'' Time

The surprising result is that our liquid crystal model system with just these three broken symmetries of life ``knows'' time, another feature of living systems. It does not learn about time after it has been pushed far from equilibrium and is close to turbulence. It knows time as soon as the nonequilibrium driving force is strong enough to enforce a pattern. Even its ®rst pattern is characterized by a frequency, o. When v ! 0, the interface reverts back to its featureless dead state: q ˆ 0 and o ˆ 0.

15.4.1

Cellular Patterns: q 0 0 and o 0 0

To recognize that there is a time scale at the pattern onset is subtle. The evidence is that the pattern travels parallel to the interface at speed vx as soon as q 0 0. At the pattern onset, vx is small (@1 mm/s) so its movement is not obvious to the impatient eye. With increasing distance from equilibrium, measured by a control parameter called e 1 …v ÿ vc †=vc , vx increases (Figure 15.2) so that its motion becomes more obvious. Despite all the drama at the cellular interface when e < 3:5 (Figure 15.2), q is linear in e : q=q0 ˆ 0:19e ‡ 0:29, independent of the dynamics. On the other hand, vx has two responses. When e < 2, vx =vel ˆ 0:32e ‡ 0:3: the cholesteric texture left by the traveling phase boundary is unstructured and there are no director relaxation processes observed when v ! 0. When e > 2, vx =vel ˆ 0:60e ‡ 0:03 : vx increases twice as fast as when e < 2. The texture left behind the traveling interface is a uniform array of line defects (one line per cell) that slowly disappears by defect coarsening when v ! 0 [1]. We note that taking into account the scatter in the data, the intercept of vx 's e > 2 line is zero and that it intersects its e < 2 line at e A 1 where the breathing mode starts. 15.4.1.1

Breathing Mode

When 0:5q0 < q < q0 , we observed a breathing mode (Figures 15.2 and 15.3). It starts at e ˆ 1. In addition to traveling, the pattern also oscillates with a frequency, o. In one half-cycle of o, alternate grooves grow while in the other half-cycle, they shrink, doubling the pattern wavelength (Figure 15.3). For the breathing mode, o=oel ˆ ÿ0:23 ‡ 1:13q=2q0 . The breathing mode is nonlinear and nondispersive [1].

15. Traveling Phase Boundaries with Broken Symmetries of Life

487

Figure 15.2. Pattern wavenumber, q, scaled by q0 and speed parallel to the interface, vx , by vel , as a function of the control parameter, e 1 …v ÿ vc †=vc .

At e ˆ 2, the interface starts to shed defects and vx doubles (Figure 15.2). Figure 15.3 is a snapshot after the start of defect shedding. The memory of the position of the defect line at the interface at the time it was shed is topologically trapped thus decorating the breathing mode for Figure 15.3. We interpret the appearance of a frequency in the cellular regime of the traveling N*±iso. phase boundary, as resulting from a competition between the nonequilibrium length scale, q, and the equilibrium length scale, q0 .

15.4.2

Lehmann E¨ect: q ˆ 0 and o 0 0

Driving the system further from equilibrium resulted in a pattern of wavelength l appearing behind the ¯at interface (Figure 15.4). In this second

488

P.E. Cladis Figure 15.3. Snapshot of the breathing mode decorated by defect lines.

regime, q ˆ 0 (the interface is ¯at) but o 0 0. The pattern behind the interface is generated by the nonequilibrium phase winding of n at the interface. Phase winding in cholesterics was ®rst observed by Otto Lehmann, the physicist father of liquid crystals, and is known as the Lehmann e¨ect. Brand and Pleiner [3] were the ®rst to account for the Lehmann e¨ect with a straightforward symmetry argument that can be used to extract a Lehmann coe½cient [2]. 15.4.2.1

Symmetry Argument for the Lehmann E¨ect

For a helix with q0 kz, n ˆ …cos j; sin j; 0† with j ˆ q0 z ‡ j0 . j is the director phase relative to a ®xed direction in the xy-plane and j0 is an arbitrary constant. The operation z ! ÿz changes q0 ! ÿq0 : in a mirror, the righthanded helix becomes a left-handed one. This symmetry allows terms in the dynamic equation [3] coupling q0 to gradients of a scalar, i.e., terms such as dj=dt @ q0 dT=dz @ nT =lT and

15. Traveling Phase Boundaries with Broken Symmetries of Life

489

Figure 15.4. When G is large, the interface is ¯at, q ˆ 0, (top left) but the director winds at the interface, o 0 0, leaving behind in the cholesteric state a striped pattern of wavelength, l. The phase winding is coherent …j0 ˆ constant† along the interface between phase jump lines where j0 jumps by p or 2p. The di¨erent symbols in the graph are for di¨erent values of G and sample thickness: G ˆ 30 K/cm (lT ˆ 33 mm) and d ˆ 37 mm; G0 ˆ 15 K/cm (lT ˆ 67 mm) and d0 ˆ 37 mm; G0 ˆ 45 K/cm (lT ˆ 22 mm) and d0 ˆ 80 mm and G ˆ 15 K/cm and d ˆ 67 mm. For a given G, the Lehmann e¨ect starts at a minimum v (e.g., left arrow for G ) and there are a few phase jump lines. As v increases, phase jump lines become more numerous (e.g., middle arrow for G ). Finally, at a maximum v for a given G, they are so numerous l can no longer be measured (e.g., right arrow for G ). We identify all points not on the universal parabola, as being in the orientational glass state.

dj=dt @ q0 dc=dz @ nc =lD . nT and nc are the Lehmann coe½cients for the coupling of q0 to temperature and concentration gradients, respectively. dj=dt 0 0 means that n rotates in time at ®xed z in the lab frame in a process we called phase winding. As lD f lT , it is the dominant e¨ect to account for phase winding in (Figure 15.4). Putting in the constants, we have g1 dj=dt ˆ nc =lD . In a ``back of the envelope'' calculation (Figure 15.4), we expand this last

490

P.E. Cladis

to relate the observed wavelength behind the interface, l, to the interface speed, v, and hence deduce the Lehmann coe½cient [2].

15.4.3

Orientational Glass

With increasing v, the spatial correlations in phase winding became shorter and shorter as the system is driven still further from equilibrium (Figure 15.4). Eventually a spatially and temporally disordered state is observed. We call this last state an orientational glass as it is characterized by many q's and many o's.

15.4.4

The Crucial RoÃle of q0

15.4.4.1

Thermal Di¨usion

The characteristic length for this process is the thermal length, de®ned by lT 1 DTc =G where DTc is the width of the two phase region of the cholesteric±isotropic transition. As G increases, lT decreases. For the concentration of C15 used in this experiment DTc ˆ 0:1 K giving an lT A 133 mm. This is about 3.5 times p0 and close to the pattern wavelength at onset (Figure 15.2). 15.4.4.2

Orientational Di¨usion

The liquid crystal's orientational di¨usion process is the dominant process at work in the cellular regime where q 0 0 and o 0 0. The di¨usion constant characterizing this process is D0 ˆ 8  10ÿ7 cm 2 /s [9]. By scaling q with q0 ˆ 2p=38 mm, vx with vel ˆ D0 = p0 ˆ 2:1 mm/s and o by oel , we saw that vx =vel , q=q0 and o=oel are O…1†. 15.4.4.3

Lehmann Coupling

Further from equilibrium, the very steep concentration gradient in advance of the moving interface triggers a Lehmann coupling to q0 . This process is characterized by the dynamic di¨usion length, lD ˆ DI =v. As v increases, lD becomes even smaller. When the Lehmann e¨ect appears (Figure 15.4), lD A 0:1 mm making impurity di¨usion the dominant e¨ect controlling the coupling between q0 and gradients in a scalar ®eld. While the interface is ¯at, the director winds at the interface leaving behind a striped pattern: q ˆ 0 and o 0 0. In these patterns, phase winding is coherent between phase jump lines where j0 jumps by p or 2p. As the interface travels faster in the temperature gradient, the distance between phase jump lines decreases as they become more numerous. Eventually, there are so many phase jump lines, a wavelength can no longer be measured and the orientational glass state takes over. In the orientational glass state, the texture behind the interface is disordered in both space (many q's) and time (many o's).

15. Traveling Phase Boundaries with Broken Symmetries of Life

491

Figure 15.5. Comparison between patterns with (bottom) and without (top) chirality.

15.5

Phase Diagram

The crucial role played by chirality in these studies is highlighted with the type of phase diagram (Figure 15.5) much beloved by Professor Heppke. The two control parameters for these experiments are v and G, both of which can be precisely controlled. Equilibrium is when v and G are zero. As v and G increase, the system moves further from equilibrium. The phase diagram for the traveling cholesteric±isotropic interface has three regimes (Figure 15.5, bottom). In the ®rst regime, where cellular patterns are observed, q 0 0 and o 0 0, the dominant di¨usion process is orientational di¨usion. The breathing mode is observed in this regime. In the second regime, where phase winding is observed, q ˆ 0 and o 0 0, the dominant process is a Lehmann coupling to q0 driven by the steep impurity gradient in advance of the traveling interface. G is su½ciently large that the interface is ¯at but there is a pattern left behind the interface. The last regime

492

P.E. Cladis

is the orientational glass state where the cholesteric left behind the traveling interface is both spatially and temporally disordered. The cholesteric phase diagram can be compared to what is observed in a nematic liquid crystal …q0 1 0† under the same nonequilibrium conditions (Figure 15.5, top). In the traveling nematic±isotropic interface, only the reentrant cellular regime near equilibrium is observed. And in this regime, there is no frequency associated with the patterns: q 0 0 but o 1 0. At higher speeds and larger temperature gradients, the interface is ¯at and dead: q ˆ 0 and o ˆ 0.

15.6

Conclusions

The ®rst surprising conclusion is that a model system with just the three elements that we have characterized as the Broken Symmetries of Life, also ``knows time.'' Its nonchiral analogue, the traveling nematic±isotropic phase boundary does not ``know time.'' The argument given is that the existence of an intrinsic length, p0 , in cholesteric liquid crystals, implies a frequency for its response to perturbations in its structure. The second surprising feature is our minimal model's novel route to turbulence. Its non-chiral analogue prepared under identical conditions has no routes to turbulence. In contrast, as it was driven further from equilibrium, our minimal model's repertoire ranges from a cellular pattern with a single wavelength and frequency, through a wavelength doubling breathing mode, followed by a phase winding ¯at interface that eventually becomes turbulent. The macroscopic implication of the Broken Symmetries of Life shown by the minimal model is profound: because living systems necessarily know time, they also have access to turbulence. Finally, we conclude that with his interest in chirality in liquid crystals for many years now, an interest we have shared, Professor Gerd Heppke has demonstrated his perspicacity and good taste in scienti®c problems. May you have many more years of happy experiences o¨ered by the ineluctable pleasures of chiralityÐand the Broken Symmetries of LifeÐparticularly broken time reversal symmetry. XroÂnia Polla0 !

References [1] P.E. Cladis, J.T. Gleeson, P.L. Finn, and H.R. Brand, Breathing mode in a pattern forming system with two competing lengths, Phys. Rev. Lett. 67, 3239 (1991); P.E. Cladis, Pattern formation at the cholesteric-isotropic interface, in: Pattern Formation in Complex Dissipative Systems (edited by S. Kai), World Scienti®c, Singapore, 1992, p. 3. [2] H.R. Brand and P.E. Cladis, Nonequilibrium phase winding and its breakdown at a chiral interface Phys. Rev. Lett. 72, 104 (1994); P.E. Cladis and H.R. Brand,

15. Traveling Phase Boundaries with Broken Symmetries of Life

[3] [4] [5] [6] [7] [8] [9]

493

Nonequilibrium phase winding and its breakdown at a chiral interface, in: SpatioTemporal Patterns in Nonequilibrium Complex Systems (edited by P.E. Cladis and P. Pal¨y-Muhoray), Addison Wesley, Reading, MA, 1995, p. 123. H.R. Brand and H. Pleiner, New theoretical results for the Lehmann e¨ect in cholesteric liquid crystals, Phys. Rev. A37, R2736 (1988). J.S. Langer, Science 243, 1150 (1989). K.A. Jackson and J.D. Hunt, Transparent compounds that freeze like metals, Acta Metall. 13, 1212 (1965). P.E. Cladis, A.J. Slaney, J.W. Goodby, and H.R. Brand, Pattern formation at the traveling liquid crystal twist grain boundary smectic A interface, Phys. Rev. Lett. 72, 226 (1994). M. Hara, H. Takezoe, and A. Fukuda, Jpn. J. Appl. Phys. 25, 1756 (1986). J.W. Rutter and B. Chalmers, Can. J. Phys. 31, 15 (1953); W.A. Tiller, K.A. Jackson, J.W. Rutter, and B. Chalmers, Acta. Metall. 1, 428 (1953). D0 ˆ K2 =g1 where K2 is the twist elastic constant measured for 8CB by M.J. Bradshaw, E.P. Raynes, J.D. Bunning, and T.E. Faber, J. Phys. (Paris) 46, 1513 (1985); and g1 ˆ 0:25 dyn/cm 2 /s by H. Kneppe, F. Schneider, and N.K. Sharma, J. Chem. Phys. 77, 3203 (1982).

This page intentionally left blank

Index

a-helix, 104 absolute con®guration, 3, 71 achiral helical twisting power (AHTP), 73 active matrix LCD, 283 adaptive laser optics, a alignment layer, 288, 289 alignment tensor, 14 amino acids, 3f, 11, 103 aminoantraquinone, 74 amphiphilic, 104, 447, 448 anchoring, 120, 137, 140±145, 149, 152±154 geometrical, 120, 121 intrinsic in cholesterics, 154 anisotropic circular dichroism (ACD), 73 ANNNI model, 269 anomalous electrostriction, 211 antiferroelectricity, 251, 274, 346, 348 asymmetric synthesis, 102, 113 atomic force microscopy, 325 atropisomers, 70 axial chirality, 108 bend grain boundary phase, 346±347 bent molecular con®guration, 263 biaxial nematics, 115, 132, 138 bicontinuous cubic phase, 104 binaphthyl, 95 birefringence, 15, 368, 408±410, 412, 416±417 bistability, 159, 166±167, 418, 421, 423± 424

blue phases, 17, 46, 102, 124±125, 186, 455 BPI, 186 BPII, 46, 186 BPIII, 186 smectic blue phase (BPS), 345±346 bond-orientational order, 199 boojum, 121, 143, 144 Bragg relation in cholesterics, 42 Bragg scattering, 200 of light (see selective re¯ection) of X-rays (see X-ray) breathing mode, 486±487, 491±492 broken symmetries of life, 481±482, 492 C2 axis, 259, 263 Cahn-Ingold-Prelog nomenclature, 72 calamitic, 13 calorimetry, 330±332, 335±338 Cano wedge, 33±37, 137, 153 carbohydrates, 3, 11 cellulose, 105, 108 chiral: axis, 70, 108 centers, 70 dopant, 101, 110 induction, 67 nematic display, 417 nematic droplets, 375, 418 nematic liquid crystal, 160, 178 nematic, 104, 186 planes, 70 pool, 102 separation techniques, 102 textures, 358±363

495

496

Index

chirality, 187, 191, 193 axial, 10 of atoms, 7 interaction tensor, 92 measurement, 68 molecular, 2¨ plane, 108 transfer, 68, 71 ch-LUD, a cholesteric, 18, 67, 71±72, 91, 101, 103, 105, 109, 118±119, 126, 128, 130, 132±134, 136, 140, 144, 147, 149, 150±151, 153±154, 159±160, 175, 186 elastomer, 434 structures, 140, 154, 159±160, 162, 168±169, 176, 178 cholesteric-nematic phase transition, 164±165, 167, 173 chromatography, 101, 113 chromonic, 109, 468, 470 chromophore, 81 circular dichroism (CD), 70, 356 circular dichroism, 433 Cladis' orbiting disclinations, 56±58 clock model, 272, 273 coarse grain model of cholesteric, 119 collective model, 275, 276, 277, 280 columnar aggregates, 109 columnar phases, 104, 109, 348, 355± 374 compression, 434±435 con®nement, 115, 116, 120, 139 conformation, 465 conformational chirality, 69 conoscopic ®gure, 267, 268, 273 conoscopic image: of the SmC* phase, 51±52 contergan, 5 Cotton e¨ect, 72 critical micellar concentration (CMC), 451 CsPFO, 453 CTAB, 453 cubic phases, 104, 192, 453 curvature, 119, 148, 152 Gaussian, 119, 120, 124, 150 principal l, 119

Darwin band in cholesterics, 42 defect theory, 196 defect-mediated melting, 188 defects. See disclination; dislocation; line defect; point defect; wall defect defect-stabilized phases, 126 de¯ection, 168 deformations and elastic constants: of bend (K3 ), 117, 118 of mixed, 117, 136 of saddle-splay (K), 119, 150 of saddle-splay (K24 ), 117, 118, 122, 123, 124, 125, 150 of splay (K1 ), 117, 118 of twist (K2 ), 117, 118 Devil's staircase, 269 diastereomer, 9 diastereomeric interaction, 74 dielectric permittivity tensor eij , 48±50 in blue phases, 48±50 in cholesterics, 41 di¨raction gratings, 155 dimer, 259, 261, 262 dino¯agellate chromosomes, 124 directional solidi®cation, 483±485 director, 14 disclinations, 37, 53, 115, 188±189, 196, 315 in a Cano wedge, 37 in the cholesteric phase, 132±139 in the cholesteric phase: l, 134±137 in the cholesteric phase: t, 134±137 in the cholesteric phase: w, 37, 132, 134±138 in the cholesteric phase: merger and splitting, 134 entanglement, 138, 139 escape in the third dimension, 131 homotopy classi®cation, 130 linked rings, 148 nonsingular, 131 singular, 131 in smectic-C* ®lms (2p), 53 strength of, 128, 131, 135 thick and thin, 118, 134 topological charge of, 132 transformations of, 131, 135, 136 in the uniaxial nematic phase, 130± 132

Index discotic, 13 dislocations, 115, 126, 130, 134±138, 150±154 dislocations: edge, 300, 347 screw, 296±300, 320, 330 dispersion interaction, 474, 475 dispiration, 257 displays: antiferroelectric liquid crystals (AFLC), 19, 226, 251 polymer-dispersed liquid crystals (PDLC), 154, 375±376, 379, 427, 429±431 polymer-stabilized cholesteric texture (PSCT), 426 surface-stabilized ferroelectric liquid crystal (SSFLC), 235f twisted nematic (TN), 15, 28, 33, 61, 161 distomer, 5 distribution function, 76 DNA, 5, 125±130, 150, 370 dopant, 459 double twist, 122±125, 196, 477 driving scheme, 284, 285 droplets, cholesteric, 142±144 division of, 144 monopole structures, 142, 143 droplets, nematic, 120±123 optical activity, 120±123 droplets: sessile, 120, 121 suspended, 121±123, 142±144 Dupin cyclides, 148 dynamic electromechanical measurement, 444 elasticity, 433 elastomer, 433 elasto-optic coe½cient, 210 electric ®eld controlled colors and light, a electric ®eld e¨ects: in TGBA phases, 339±343 in TGBC phases, 343±345 electroclinic e¨ect, 102, 339±342 in¯uence of enantiomeric excess, 244 smectic-A phase, 223, 236±241 electromechanical: behavior, 438 measurement, 441 response, 437, 445

497

electron microscopy, 322±325 electro-optic application, 15. See also displays electro-optic e¨ects: antiferroelectric switching, 245 DHF-e¨ect, 237 electroclinic e¨ect, 19, 102, 223, 226, 236±237, 239, 241, 244, 245, 339, 366 electrostriction of blue phases, 209, 210 ferroelectric switching (SmC* phase), 235f ¯exoelectric switching, 121 Frederiks transition, 121 TGB phases, 21, 139, 296, 301, 306± 310, 313±315, 318, 320, 325, 331± 333, 337, 339, 345±348 twisted nematic (TN) display, 28, 33, 60, 161 V-shaped switching, 274, 287 electro-optical switching, 365±370 electrostriction coe½cient, 210 electrostriction, 209 ellipsometry, 256, 267 enantiomer, 5, 7, 9, 69, 191 enantiomeric excess (ee), 69 in¯uence on electroclinic e¨ect, 244 in¯uence on ferroelectric properties, 243 enantiomeric purity, 107 enzymes, 4, 11 Ericksen stress, 58 escape in the third dimension, 131, 132, 133 Euler characteristic E, 119, 129, 140, 144 eutomer, 5 exciton coupling, 81 fatty acids, 104 ferrielectric, 251, 274, 368 ferroelectric, 18f, 102, 105, 251, 274, 368, 433, 438, 455, 474 ferroelectric properties: elastomers, 433, 438 in¯uence of enantiomeric excess, 243 of nonchiral compounds, 245 smectic-C phase, 223¨ TGB phases, 341±345

498

Index

ferro¯uid, 457 ®ngerprint texture, 138, 455, 457 ®ngers: in cholesterics, 144, 145, 146 Fischer projection, 3 ¯exoelectric polarization, 227 focal conic: domains, 115, 119, 148, 149, 150 elastic vs. anchoring energy, 149 forced phase winding, 483, 488±493 Fredericks transition, 31, 121 free elastic energy, of a cholesteric, 117 coarse grain, 328 free-standing ®lms of the SmC* phase, 51±58 freeze fracture, 322±325 frustration, 188, 274, 283 full pitch band, 255 Gauss theorem, 141 Gauss-Bonnet theorem, 119, 120 glycolipids, 104 Goldstone mode, 234f Grandjean-Cano-lines: in the cholesteric phase, 33±37, 137, 153 in TGB-phases, 315±317 Hausdor¨ distance, 478 heat capacity, 190 helical (twisted) texture, 29 in a twisted nematic display, 31 in the cholesteric phase, 34 helical inversion, 102, 112 helical pitch. See pitch helical structure: smectic-C phase, 226 helical structures, 13, 16, 20, 186 lyotropic, 13 nonlinear optics, 20 thermotropic, 13 helical twisting power (HTP), 67, 110± 112, 462 helical unwinding: of blue phases (see electrostriction) of the cholesteric phase, 40, 41, 44, 61, 66, 110, 115, 297, 299, 301, 319, 433 of smectic phases, 339 of TGB phases, 339 helicity rules, 70

helicity tensor, 92 helicoidal state, 254 helix, 4f, 24f inversion, 86, 464 herring bone, 251, 252, 255, 256 hexagonal packing and twist, 125, 126 hexagonal, 453 homotopy classi®cation, 130 homotopy groups, ®rst (fundamental), 130 non-Abelian, 133, 138, 139 relative, 146 homotopy groups, second, 130 homtopy theory, 126 Hopf invariant, 148 Hopf textures, 147 hydrogen bond, 450, 470 hydrophobic e¨ect, 450 hysteresis, 252, 253 inherent dissymmetric, 70 insulating layer, 288, 289 intermolecular chirality transfer, 68 intramolecular chirality transfer, 68 inversion, 102, 112. See also pitch inversion; helical inversion IR (polarized), 261, 263, 277, 279 Ising model, 268 ITO electrodes: in a twisted nematic display, 29 Kossel diagrams of blue phases, 44±51, 202 Kossel rings, 50±51 lamellar, 104, 453 Landau free energy, 192 Landau theory, 192, 232¨, 238. See also SmA-SmC transition lasing, a layer dilatation, 119 LCD. See displays left-handed and right-handed cholesteric, 117 Lehmann e¨ect, 487±488, 490, 493 levels of chirality, 71 light scattering by the cholesteric texture, 42

Index line defects, 116, 118, 130, 131, 188, 389, 390±391, 396, 398, 405, 486 linking number, 128, 129 lipids, 104 liquid crystal display (LCD), 283, 287, 289, 290, 291. See also displays liquid crystal induced CD (LCICD), 73 liquid crystal network, 433 living systems, 24 long-range orientational order, 67 long-range positional order, 67 lyotropic blue phases, 109 lyotropic cholesteric phase, 104 lyotropic phases, 104±109, 348 magnetic coherence length, 38 Maxwell equations: in cholesterics, 62± 66 mechanical ®eld, 438 mesoform, 9 mesophase chirality, 101, 102 micelle, 104, 108, 109, 451 MoÈbius strip, 127, 128 model of cholesteric, 119 modes of light propagation: in cholesterics, 61±66 in twisted texture, 32, 59±61 Moire state, 363, 364 molecular chirality, 101 monopole structures, 142, 143, 150 Monte Carlo simulation, 91 nano-structures, 109 natural polymers, 108 natural systems, 370±374 necklace, 474 nematic eutectic mixture, 31 nematic prism, 28 network (crosslinked polymer), 433 neutrino, 5 NMR, 113 nonequilibrium physics, a nonlinear optic, 433 nonlinear optics, 27, 171, 173, 184±185, 433 nonlinear physics, a odd-even behavior, 111, 258, 259 oily streaks, 119, 148, 150±153

499

buckling in ®eld, 152, 153 optical activity, 1f, 120 optical anisotropy, 103, 275. See also birefringence optical purity, 267, 271. See also enantiomeric purity optical rotatory dispersion (ORD), 79 optical switching, 178±179 order parameter space, 130±133, 144, 147 order parameter, 78, 110, 192 order triangle, 76 orientation axis, 85 orientational distribution coe½cients, 76 orientational distribution function, 76 orientational glass, 490, 492 orientational order, 200 packing parameter, 451 passive matrix LCD, 283 persistence length, 118 phase diagram, 192 photo-induced change of pitch, 172 photoisomerization, 101 piezoelectric, 368, 434 pitch inversion, 112. See also helical inversion pitch, 102, 110, 188 cholesteric phase, 34, 112 electrically induced change (see helical unwinding) photo-induced change (see photoinduced) smectic phases, 18, 223, 228, 296 TGB phases, 318±319 pixel: in a twisted nematic display, 29± 33 planar anchoring, 34, 379, 385, 393, 398 in a Cano wedge, 34 in a twisted nematic display, 30 planar chirality, 108 Poincare theorem, 141 point defects, 130, 149 boojums, 121, 143, 144 hedgehogs, 141, 145 homotopy classi®cation, 130 topological charges, 141 polar order, 281 polarization charge, 282

500

Index

poly-g-benzylglutamate (PBG), 121, 125 polygonal textures, 149 polymer dispersed liquid crystals, 169, 249 polymer network, 433 polymers stabilized liquid crystals, 375 polymers, 108, 348 polypeptides, 125, 150 polysaccarides, 125 positional distribution function, 76 positional order, 200 pressure, 325±326 pretransitional e¨ect, 285 Prorocentrum Micans, 124 pseudoscalar, 70 pseudotensor, 97 Px model, 259 pyroelectric, 357, 358, 433, 440 racemate, 191 racemic separation, 113 racemic-achiral, 68 random model, 277 reentrant, 265 re¯ection microscopy, 190 refractive index, 15, 42, 162, 172, 174, 178±179, 201, 212, 281, 455 response time, 285, 286, 287 rotamers, 69 rotating ®eld e¨ects: in SmC* ®lms, 56± 58 rotatory power, 205 rubber elasticity, 433 second-harmonic generation, 275, 279 sector rule, 70 selection rules for light propagation: in cholesterics, 43 selective re¯ection, 16f, 201, 255, 317± 320 of blue phases, 17, 190±191, 200 of cholesteric phases, 39±44 of smectic phases, 296, 366 of TGB phases, 339 sensors, 101, 113 SHG interferometry, 281±282 SHG, 279 slab thickness: in TGBA , 297, 320, 325 in TGBC , 297, 328, 330±333, 344±345

SmA±SmC transition: in¯uence of enantiomeric excess, 241 Landau theory, 232¨ tilt angle, 225 smart mirrors, 155 smectic phases, 224f smectic-A±smectic-C transition: see SmA±SmC smectic-A phase (SmA), 105, 296±301, 339 electroclinic e¨ect, 223, 236±241 structure, 224¨ smectic-Cg* phase (SmCg*), 265, 266, 267, 269±273 smectic-Ca* phase (SmCa*), 102, 265, 273, 274 smectic-C* elastomer, 438 smectic-C* phase (SmC*), 342±345 dielectric properties, 234f electro-optic e¨ects, 235f ferroelectric properties, 51±53, 223¨ ¯exoelectric polarization, 227 helical structure, 226 soft mode, 234f, 238 structure, 51, 224¨ symmetry, 228 tilt angle, 224f smectic-CA …† phase (SmCA …† ), 256 smectic-O phase (SmO), 256, 346 smectic-Q phase (SmQ), 346 space group, 195 speci®c ratation: ampli®cation of, 362 spherical harmonic, 193 spherulites, 125, 144, 146 Robinson, 150 spiral textures, 358±363 splay-bend (K13 ), 117, 118 spontaneous polarization, 110±112 coupling to tilt angle, 231 in¯uence of enantiomeric excess, 243 Landau theory, 232¨ molecular origin, 228 sign, 230f SmC* phase, 51±53 TGB phases, 326±327, 339±345 SSFLC cell, 235f stereoselective synthesis, 10¨ sterical hindrance, 103, 468

Index sterical interaction, 474 steroids, 102, 103, 112 stripe structures, 121 structure of smectic phases, 224¨ subphase, 263, 268 sugars, 104, 105 supermolecular structures, 24 suprastructural chirality, 67 surfactant, 448 switchable chirality, 112 symmetry of smectic-C phase, 228 symmetry operations: of the ®rst kind, 68 of the second kind Sn, 68 TADDOL's, 87 tartaric acid, 1, 9 terpenoids, 103 texture: in the Cano wedge, 35 thermotropic, 108 amphiphilic systems, 109 threshold, 252, 253, 284, 287 thresholdless, 274 threshold voltage: for the Fredericks transition in TND, 31 tilt angle: of smectic-C phase, 224f tilt grain boundary phases, 364, 365 topological charge, 126±128, 141 illustration with closed stripes, 126± 128 topological con®guration (soliton), 130, 146±148 linear soliton, 146 particle soliton, 147, 148 planar solition, 146 topological defects: homotopy classi®cation, 130

501

topological ¯ows: in SmC* ®lms, 58 traveling phase transition boundaries, 481 tristable switching, 252, 257, 283 tunable chirality, 112 twin compound, 105 twist, 129 twist grain boundary phases, 18, 102, 126, 139 compounds, 308±314 TGB2q , 296, 334±335 TGBA , 102, 296±326, 336, 339±343 TGBC , 102, 296, 326±333, 336, 343±345 TGBC  , 296, 333±334, 348 twisted line liquid, 296, 335 twisted nematic (TN) display, 15, 19, 28±33 undulated TGB phase, 333±334 unwinding: of the cholesteric phase, 37±39 viscoelasticity, 205 viscosity, 103 Volterra process, 136 V-shaped switching, 274, 287 wall defects: of p and 2p in SmC* ®lms, 55 weak interaction, 6¨ writhe Wr, 129 xantan, 125 X-ray di¨raction: TGBA , 320±322 TGBC , 328±332 XY model, 268